Twisted submanifolds of R^n

We propose a general procedure to construct noncommutative deformations of an embedded submanifold $M$ of $\mathbb{R}^n$ determined by a set of smooth equations $f^a(x)=0$. We use the framework of Drinfel'd twist deformation of differential geometry of [Aschieri et al., Class. Quantum Gravity 23 (2006), 1883]; the commutative pointwise product is replaced by a (generally noncommutative) $\star$-product determined by a Drinfel'd twist. The twists we employ are based on the Lie algebra $\Xi_t$ of vector fields that are tangent to all the submanifolds that are level sets of the $f^a$; the twisted Cartan calculus is automatically equivariant under twisted tangent infinitesimal diffeomorphisms. We can consistently project a connection from the twisted $\mathbb{R}^n$ to the twisted $M$ if the twist is based on a suitable Lie subalgebra $\mathfrak{e}\subset\Xi_t$. If we endow $\mathbb{R}^n$ with a metric then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi-Civita connection consistently to the twisted $M$, provided the twist is based on the Lie subalgebra $\mathfrak{k}\subset\mathfrak{e}$ of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean $\mathbb{R}^3$ and twisted hyperboloids embedded in twisted Minkowski $\mathbb{R}^3$ [these are twisted (anti-)de Sitter spaces $dS_2,AdS_2$].


Introduction
The notion of a submanifold N of a manifold M is a fundamental concept in (differential) geometry, playing a crucial role in various branches of mathematics and physics. A metric, connection, ..., on M uniquely induces (see e.g. [40]) a metric, connection, ..., on N . In the last few decades various deep physical and mathematical reasons have stimulated the generalization of differential geometry to so-called Noncommutative Geometry (NCG) [13,42,44,45,38]. In particular, NCG has been advocated as a suitable framework for formulating a fundamental (or at least an effective) theory of quantum spacetime allowing the quantization of gravity (see e.g. [18,1]) and/or for unifying fundamental interactions (see e.g. [14,11]). It is therefore natural and important to investigate whether and to what extent a notion of a submanifold is possible and fruitful in the NCG framework. Surprisingly, this question has received little systematic attention so far (rather isolated exceptions are the papers [47,37,54]). On several noncommutative (NC) spaces one can make sense of special classes of NC submanifolds, but some features of the latter may depart from their commutative counterparts. For instance, from the noncommutative algebra "of functions on the quantum group U q (n)", which is generated by the n 2 matrix elements of a n × n unitary matrix, one can obtain the one A on the quantum group SU q (n) by imposing that the so-called q-determinant (a suitable central element) be 1, as in the commutative (q = 1) limit; but the so-called quantum group bicovariant differential calculus on A (i.e. the corresponding A-bimodule Ω of 1-forms) remains of dimension n 2 instead of n 2 − 1 [56]. The same phenomenon occurs e.g. obtaining the SO q (n)-covariant quantum Euclidean spheres S n−1 q from the SO q (n)-covariant quantum Euclidean spaces R n q , by imposing that the [central and SO q (n)-invariant] square distance r 2 from the origin be 1; said differently, the 1-form dr 2 cannot be imposed to vanish, and actually the graded commutator 1 q 2 −1 r −2 dr 2 , · acts as the exterior derivative [31,52,28,10].
In the present work we wish to address the above question systematically within the framework of deformation quantization [6] in the particular approach based on Drinfel'd twisting [19] of Hopf algebras. We restrict our attention to the noncommutative generalization of embedded submanifolds of R n , because by the Whitney embedding theorems [43] one can always embed a smooth manifold M in R n with a sufficiently high dimension n. More precisely, we shall assume that M ⊂ R n consists of points of x ∈ R n fulfilling a set of equations f a (x) = 0, a = 1, 2, ..., k < n, where f ≡ (f 1 , ..., f k ) : R n → R k are smooth functions such that the Jacobian matrix J = ∂f /∂x is of rank k on all R n ; or, more generally, that f is well-defined and J is of rank k on an open subset D f ⊂ R n , and M consists of the points of D f fulfilling (1). In all our examples here E f := R n \ D f will be empty or of zero measure. By replacing in (1) f a (x) → f a c (x) := f a (x) − c a , with c ≡ (c 1 , ..., c k ) ∈ f (D f ), we obtain a whole k-parameter family of embedded manifolds M c (M 0 = M ) of dimension n − k, that are level sets of f . Embedded submanifolds N ⊂ M can be obtained by adding more equations to (1) 1 . The *algebra X M of smooth complex-valued functions on M can be expressed as the quotient of the * -algebra X = C ∞ (D f ) of smooth functions on D f over the ideal C ⊂ X of smooth functions vanishing on M : In the appendix we prove that C is generated by the left-hand sides (lhs) of (1): Theorem 1 C = k a=1 X f a = k a=1 f a X , i.e. for all h ∈ C there exist h a ∈ X such that Similarly, X N can be obtained as the quotient of X M over the ideal generated by further equations of the type (1), or equivalently as the quotient of X over the larger ideal generated by all such equations. Identifying vector fields with first order differential operators, we denote as Ξ := {X = X i ∂ i | X i ∈ X } the Lie algebra of smooth vector fields X on D f ; here and below we abbreviate ∂ i ≡ ∂/∂x i . Those vector fields X ∈ Ξ such that X(f a ) belong to C for all a, or equivalently such that X(h) belongs to C if h does (i.e. vanishes when restricted to M ) make up a Lie subalgebra Ξ C , which is also a X -bimodule; those such that X(h) belongs to C for all h ∈ X make up a smaller Lie subalgebra Ξ CC , which is actually an ideal in Ξ C and itself a X -bimodule. It decomposes as Ξ CC = k a=1 f a Ξ. The Lie algebra Ξ M of vector fields tangent to M can be identified with that of derivations of X M , namely with the quotient If f a (x) are polynomial functions fulfilling suitable irreducibility conditions and we set X = Pol • (R n ), the * -algebra of complex-valued polynomial functions on R n (instead of X = C ∞ (D f )), then again the * -algebra X M of complex-valued polynomial functions on M can be expressed as the quotient X M = X /C, where C ⊂ X is the ideal of polynomial functions vanishing on M , Ξ := {X = X i ∂ i | X i ∈ X } is the Lie algebra of polynomial vector fields X on R n , etc. C can be decomposed again in the form (3), with X = Pol • (R n ) [35].
Often one is interested in noncommutative deformations of differential geometry on a manifold, i.e. in families of NCGs depending on a formal parameter ν and reducing to the original one if we formally set ν = 0. Deformation quantization [6,41] provides a general framework to deform X into a noncommutative algebra X over C [[ν]] (the ring of formal power series in ν with coefficients in C): as a module over C [[ν]] X coincides with X [[ν]], but the commutative pointwise product αβ of α, β ∈ X (C[[ν]]-bilinearly extended to X [[ν]]) is deformed into a possibly noncommutative (but still associative) product, where B l are suitable bidifferential operators of degree l at most. We wish to deform X M into a noncommutative algebra X M in the form of a quotient with C a two-sided ideal of X . To make also X M = X for all α ∈ X , a = 1, .., k [ (7) implies that the f a are central in X , again]. In [19] Drinfel'd has introduced a general deformation quantization procedure of universal enveloping algebras U g (seen as Hopf algebras) of Lie groups G and of their module algebras, based on twisting; a twist is a suitable element (a 2-cocycle, see section 2.1.1) 2 In fact, for all c ≡ k a=1 f a c a ∈ C (c a ∈ X ) (7) implies c = k a=1 f a c a , so that for all α ∈ X , by the associativity of , c α = ( k a=1 f a c a ) α = k a=1 f a (c a α) (7) = k a=1 f a (c a α) ∈ C[[ν]]; and similarly for α c. Note that it is not sufficient to require that α f a −αf a , f a α−f a α, or equivalently B l (α, f a ), B l (f a , α), belong to C to obtain the same results. As a more general condition ensuring c α, α c ∈ C[[ν]] one could require that for all a = 1, .., k and α ∈ X the product f a α = αf a can be expressed as a combination of -products: ). It acts on the tensor product of any two U g-modules or module algebras, in particular algebras of functions on any G-manifold, including some symplectic 3 manifolds [4].
In [1] the authors consider the Lie algebra g = Ξ M of smooth vector fields on a generic smooth manifold M (this is the Lie algebra of the infinite-dimensional Lie group of diffeomorphisms of M ) and the U Ξ M -module algebra X M = C ∞ (M ); F I l 1 , F I l 2 seen as differential operators acting on X M have order l at most and no zero-order term. The deformed product reads where F = 1⊗1 + ∞ l=1 ν l I l F is the inverse of the twist. In the sequel we will abbreviate F = F 1 ⊗ F 2 , F = F 1 ⊗ F 2 (Sweedler notation with suppressed summation symbols). In the presence of several copies of F we write F 1 ⊗F 2 and F 1 ⊗F 2 etc., in order to distinguish the summations. Actually, Ref. [1] twists not only U Ξ M into a new Hopf algebra U Ξ F M and X M into a U Ξ F M -module algebra X M , but also the U Ξ M -equivariant X M -bimodule of differential forms on M , their tensor powers, the Lie derivative, and the geometry on M (metric, connection, curvature, torsion,...) -if present -, into deformed counterparts.
Here, as in [47], we shall take the algebraic characterization (2)(3)(4) as the starting point for defining submanifolds in NCG, but use a twist-deformed differential calculus on it 4 .
is the Lie subalgebra of vector fields tangent to all submanifolds M c (because they fulfill X(f a c ) = 0 for all c ∈ f (D f )) at all points; it is an X -bimodule, as well. The key observation is that, applying this deformation procedure to X = C ∞ (D f ) with a twist F ∈ U Ξ t ⊗ U Ξ t [[ν]], we satisfy (7) and therefore obtain a deformation X of X such that X In other words, we obtain a noncommutative deformation, in the sense of deformation quantization and in the form of quotients as in (2)(3)(4), of the k-parameter family of embedded manifolds M c ⊂ R n . Actually, for every X ∈ Ξ C there is an element in the equivalence class [X] that belongs to Ξ t , namely its tangent projection X t (see Proposition 6) If F is unitary or real, then U Ξ F and X admit * -structures (involutions), making them a Hopf * -algebra and a U Ξ F -module *algebra, respectively; thereby U Ξ F t is a Hopf * -subalgebra and X Mc , Ξ t , ... are a U Ξ F t -module * -algebra and U Ξ F t -equivariant Lie * -algebras, respectively. By the same procedure one can obtain noncommutative deformations of differential geometry on submanifolds N ⊂ M ⊂ R n .
The plan of the paper will be as follows.
In section 2 we present preliminary material, first on twisting (section 2.1), then on its application [1,2,3] to the differential geometry on a generic manifold (section 2.2). 3 However this quantization procedure does not apply to every Poisson manifold: there are several symplectic manifolds, e.g. the symplectic 2-sphere and the symplectic Riemann surfaces of genus g > 1, which do not admit a -product induced by a Drinfel'd twist (c.f. [7,15]). Nevertheless, if one is not taking into account the Poisson structure, every G-manifold (i.e. smooth manifolds G acts on) can be quantized via the above approach. 4 The derivation-based approach to differential calculi of Dubois-Violette and Michor [20], which was used in [47], does not encompass several differential calculi (e.g. quantum group covariant ones), or requires algebra extensions to succed (see e.g. [10]). The approaches to the differential calculusà la Connes [13] and Woronowicz [56] (which include the one considered here) are more general: the bimodule of noncommutative differential 1-forms is the primary object whereby the whole calculus can be derived by imposing the Leibniz rule and nilpotency of the exterior derivative. As a result, the dual module consists of noncommutative vector fields which are no longer derivations.
In section 3 we deal with twist deformations of embedded manifolds M ⊂ R n in the smooth context. In section 3.1 we pave the way for these deformations recalling or deriving basic facts about differential geometry on a submanifold M of R n , i.e. how the Cartan calculus and any connection, metric, etc., on R n induces corresponding data on M , how to concretely build bases of the bimodules Ξ t , Ξ ⊥ of tangent and normal vectors fields (i.e. sections in the tangent and normal bundle), the corresponding projections pr t , pr ⊥ , etc. In section 3.2 we first show that the whole twisted Cartan calculus on X is projected to the one on X M , in the same way as for its undeformed counterpart, and that projection commutes with twisting, for all ]. Then we show that the same can be done for: i) a connection ∇, using a twist F ∈ U e ⊗ U e[[ν]], where e is the corresponding equivariance Lie algebra (a Lie subalgebra of Ξ t ); ii) the metric, and the associated Levi-Civita connection, using a twist F ∈ U k ⊗ U k[[ν]], where k ⊆ e is the Lie subalgebra of the corresponding Killing vector fields. Under the latter assumptions one can build a twisted version not only of the first, but also of the second fundamental form, and prove a twisted version of Gauss theorem. Twisted Ξ t , Ξ ⊥ , pr t , pr ⊥ stay essentially undeformed; we find suitable k-invariant bases for them. Here we limit ourselves to developing (pseudo)Riemannian geometry for our physical interests, but other geometric structures (say projective, affine, conformal,...) could be explored as well.
To build concrete examples of twisted submanifolds one can look for M ⊂ R n such that Ξ t contains a finite-dimensional Lie subalgebra g, because the simplest known Drinfel'd twists are based on such a g; a nontrivial g surely exists if M is symmetric under some Lie group.
In particular, one can apply [35] this procedure to algebraic submanifolds M ⊂ R n , e.g. quadrics (i.e. level sets of a polynomial function f (x) = 0 of degree 2); for the latter there exists a Lie subalgebra g (of dimension at least 2) of both Ξ t and the Lie algebra aff(n) of the affine group Aff(R n ) = R n ×GL(n) of R n . If we choose a twist F ∈ U g ⊗ U g[[ν]] all the structures can be formulated in terms of generators and -polynomial relations. More precisely, the algebra X = Pol • (R n ) of polynomial functions (with complex coefficients) in the set of Cartesian coordinates x 1 , ..., x n is deformed so that every -polynomial of degree k in x equals an ordinary polynomial of the same degree in x, and vice versa. The same occurs with the X -bimodules and algebras Ω • of differential forms, that of differential operators, etc. In [35] the authors discuss in detail deformations of all families of quadric surfaces embedded in R 3 that are induced by twists of the abelian [51] or Jordanian [49,50] type. In section 4 of the present work, as illustrations of the approach, we just present cocycle twist deformations of elliptic (in particular, circular) cylinders (first family) as well as hyperboloids and cone (second family) embedded in R 3 ; they are induced by unitary abelian or Jordanian twists. Endowing R 3 with the Euclidean (resp. Minkowski) metric gives the circular cylinders (resp. hyperboloids and cone) a Lie algebra k of isometries of dimension at least 2; choosing a twist F ∈ U k ⊗ U k[[ν]] we thus find twisted (pseudo)Riemannian M c (with the metric given by the twisted first fundamental form) that are symmetric under the Hopf algebra U k F (the "quantum group of isometries"), and the twisted Levi-Civita connection on all M c equals the projection of the twisted Levi-Civita connection on R 3 (the exterior derivative), while the twisted curvature can be expressed in terms of the twisted second fundamental form through the twisted Gauss theorem. Actually, the metric, Levi-Civita connection, intrinsic and extrinsic curvatures of the circular cylinders and hyperboloids, as elements in the appropriate tensor spaces, remain undeformed; the twist enters only their action on twisted tensor products of vector fields. The twisted hyperboloids can be seen as twisted (anti-)de Sitter spaces dS 2 , AdS 2 .
In section 5 we summarize our results, add further remarks, mention possible mathematical developments, physical applications, issues worth further investigations. As known, the universal enveloping algebra (UEA) H := U g of the Lie algebra g of any Lie group G is a Hopf algebra. First, we briefly recall what this means. Let ε, ∆ are extended to all of H as algebra maps, S as an antialgebra map: The extensions of ε, ∆, S are unambiguous, because ε(g) = 0, ∆ [g, g ] = ∆(g), ∆(g ) , The maps ε, ∆, S are the abstract operations by which one constructs the trivial representation, the tensor product of any two representations and the contragredient of any representation, respectively. H = U g equipped with ε, ∆, S is a Hopf algebra; this means that a number of properties (see e.g. [12,45,23]) are fulfilled, in particular If G is a real form of a Lie group then there exists also a * -structure on H = U g, i.e. an involution * : H → H such that for all a, b ∈ H and α, β ∈ C H equipped with * , ε, ∆, S is a Hopf * -algebra.
Secondly, we recall how to deform a Hopf algebra using a twist [19] (see also [53,12]). Let Extending the product, ∆, ε, S linearly to the formal power series in ν and setting one finds that the analogs of conditions (11), as well as analogs of the coassociativity, counitality and antipode property are satisfied and therefore H F = (Ĥ, ∆ F , ε, S F ) is a Hopf algebra deformation of (H, ∆, ε, S). While the latter was cocommutative, H F is triangular noncocommutative (or quasi-cocommutative), i.e. τ •∆ F (a) = R∆ F (a)R, where R = R 21 is the inverse of the so-called universal R-matrix or triangular structure R. Correspondingly, ∆ F , S F replace ∆, S in the tensor product of any two representations and the contragradient of any representation, respectively. Drinfel'd has shown [19] that any triangular deformation of the Hopf algebra H can be obtained in this way (up to isomorphisms).
Eq. (15), (17) imply the generalized intertwining relation ∆ reduce to ∆ F , ∆, F for n = 2, whereas for n > 2 they can be defined recursively as The result for ∆ (n) F , F n are the same if in definitions (19) we iterate the coproduct on a different sequence of tensor factors [coassociativity of ∆ F ; this follows from the coassociativity of ∆ and the cocycle condition (15)]; for instance, for n = 3 this amounts to (15) and ∆ We consider the following examples of twists: i.) Let n ∈ N, P := n i=1 e i ⊗e n+i ∈ g⊗g, with pairwise commuting elements e 1 , ..., e 2n ∈ g.
is a Drinfel'd twist on U g ( [51]). We refer to it as an abelian (or Reshetikhin) twist on U g. It is unitary if P * ⊗ * = P ; this is e.g. the case if the e i are anti-Hermitian or Hermitian. It is immediate to check that the twist with P replaced by P = 1 2 n i=1 (e i ⊗e n+i −e n+i ⊗e i ) is both unitary and real, leads to the same R and makes β = 1, whence S F = S, and the * -structure remains undeformed also for H- * -modules and module algebras, see (22).
ii.) Let H, E ∈ g be elements of a Lie algebra such that [H, E] = 2E. Then defines a Drinfel'd twist, called Jordanian twist [49,50]. If H and E are anti-Hermitian, F is unitary. More sophisticated twists can be obtained using this as a prototype [8,48,9].
There are numerous other examples of Drinfel'd twists. We refer to [22] for an explicit (recursive) construction of twists on UEA via a Fedosov method and a classification (of equivalence classes) of twists in terms of the Chevalley-Eilenberg cohomology of the Lie algebra.

Twisting H-modules and H-module algebras
We recall that, given a Hopf ( * -)algebra H over C, a left H-module (M, ) is a vector space M over C equipped with a left action, i.e. a C-bilinear map (g, a) ∈ H ×M → g a ∈ M such that (20) 1 and 1 a = a hold. An element a ∈ M of a left H-module is said to be H-invariant if g a = (g)a for all g ∈ H. Equipped also with an antilinear involution * fulfilling (20) 2 , (M, , * ) is a left H- * -module. A left H-module ( * -)algebra is a ( * -)algebra A over C equipped with a left H-( * -)module structure, such that (20) 3 and g 1 = (g)1 hold: (g a) * = [S(g)] * a * , g (ab) = g (1) a g (2) b .
If g ∈ g, formula (20) 3 becomes the Leibniz rule. An A-bimodule M for a left H-module algebra A is said to be an Given an H-module ( * -)algebra A and choosing M = A, the twist gives also a systematic way to make A[[ν]] into an H F -module ( * -)algebra A , by endowing it with a new product, the so called -product. In fact, is associative by (15), fulfills (a a ) * = a * a * and g (a a ) = g (1) a g (2) a .
If aa = ±a a, i.e. a, a (anti)commute, then Consequently, twists leading to the same R where a ∈ A and s ∈ M, structure M as an H F -equivariant A -( * -)bimodule M (with * -involution (22) on M ). We refer to [5,32] for proofs of the previous statements.
Given two H-modules (M, ), (N , ), the tensor product (M ⊗ N , ) is an H-module if we define g (a⊗b) := g (1) a ⊗ g (2) b . As above, this is extended to an H F -( * -)module (M ⊗ N [[ν]], ). Introducing the " -tensor product" [1] Given two H-module ( * -)algebras A, B, this applies in particular to M = A, N = B. The tensor ( * -)algebra A⊗B [whose product is defined by (a⊗b)(a ⊗b ) = (aa ⊗bb )] is an H-module ( * -)algebra under the action . By introducing the -product (23) A⊗B is deformed into an H F -module ( * -)algebra (A⊗B) , with product and * -structure related to those of A , B by where R 1 ⊗R 2 (again a summation is understood) is the decomposition of R in H F ⊗H F . From (29) we recognize that (A⊗B) is isomorphic to the braided tensor product (algebra) [45,12] of A with B ; here the braiding is involutive and therefore spurious, as R = R 21 . So (A⊗B) encodes both the usual -product within A, B and the -tensor product (or braided tensor product) between the two. (On the contrary, setting (a ⊗ b) := F 1 a ⊗ F 2 b 'unbraids' the braided tensor product, cf. [29]). By (15) the -tensor product is associative, and the previous results hold also for iterated -tensor products.  [39,1] (cf. also [25,32]) Namely, D(ξ ξ ) = D(ξ)D(ξ ), and ∆ , S , R are related to ∆ F , S F , R by the relations

Twisted differential geometry
Ref. [1] applies the previous machinery to H = U Ξ, where Ξ is the Lie algebra of the Lie group of diffeomorphisms of M , and A is the algebra X = C ∞ (M ) of smooth functions on M , or more generally an X -bimodule of tensor fields on M . Tensor fields of rank (p, r) (p, r ∈ N 0 ) on M can be described as elements in the tensor product of the X -bimodules Ω ≡ Ω 1 , Ξ of differential 1-forms and vector fields on M , respectively. Here and below ⊗ stands for ⊗ X (rather than ⊗ C ), namely We set T 0,0 := X . The tensor product is associative; to avoid the need of reorderings we multiply T ∈ T p,r by 1-form tensor factors only from the left if r > 0, by vector field tensor factors only from the right if p > 0. The tensor product between a function f ∈ X ≡ T 0,0 and another tensor field is as usual not explicitly written. All T p,r are X -bimodules, e.g.
where L is the Lie derivative. It fulfills the Leibniz rule g (T ⊗ T ) = g (1) T ⊗ g (2) T . By setting A = T we can apply the results of section 2.1, in particular define a deformed tensor algebra T with associative -tensor product defined by eq. (27). T is a U Ξ F -module algebra. All T h,r are X -bimodules. In T we have in particular The first formula shows that ⊗ is actually ⊗ X , the tensor product over X . While the usual product of a tensor field T with a function h from the left and from the right coincide 5 , in general this no longer occurs with the -product.
In a chart U with coordinates x µ any vector field X can be expressed in the ∂ µ basis as X = X µ ∂ µ . It can be also uniquely expressed as X = X µ ∂ µ , where X µ are functions defined on U . The same occurs if {∂ µ } is replaced by a more general (not necessarily holonomic or ν-independent) frame {e a }: X = X a e a . Similarly, every 1-form ω can be uniquely written as ω = ω µ dx µ = ω µ dx µ , with ω µ , ω µ functions defined on U , and where {dx µ } is the usual dual frame of the vector field frame {∂ µ }. More generally, in U every tensor field T p,q ∈ T p,q can be uniquely written using functions T λ 1 ...λq µ 1 ...µp defined on U as Let us twist the algebra A = Ω • = ⊕ p Ω p of differential forms. We denote by Ω • := (Ω • , ∧ ) the C[[ν]]-module of forms equipped with the -deformed wedge product This can be seen as the tensor subspace of totally -antisymmetric (contravariant) tensor fields. The degree of the top form stays undeformed. Below we drop the symbols ∧, ∧ . The usual exterior derivative d : Ω • → Ω •+1 satisfies the graded Leibniz rule d(α p β) = dα p β + (−1) p α p dβ and is therefore also the -exterior derivative. This is so, because the exterior derivative commutes with the Lie derivative, i.e. with the Hopf algebra action.
One can endow [1] the module underlying the algebra ] itself with theproduct; the new algebra U Ξ endowed by suitable coproduct, counit, antipode becomes a Hopf algebra isomorphic to U Ξ F , whereby it is manifest that the above differential calculus is bicovariant in the sense of Woronowicz [56]. Ξ is closed under the -Lie bracket The action L X of U Ξ on T ( -Lie derivative) is defined by

-Pairing between 1-forms and vector fields, twisted Cartan calculus
Denoting , the commutative pairing between vector fields and 1-forms, the -pairing is defined as , The -pairing is actually a map , : Ξ ⊗ Ω → X , as it satisfies the X -linearity properties with h, k ∈ X . From X, dh = X(h), g dh = d(g h) and (40) it follows that X is a twisted derivation, i.e. fulfills the deformed Leibniz rule the quickest way to prove the latter is by the Leibniz rule for d and (42), (41), (25). The compatibility X Y, ω = X (1) Y, X (2) ω of , with the Lie derivative (which expresses the diffeomorphism-invariance of the pairing) implies In the commutative case, for any local moving frame (vielbein) {e i } we can build a dual frame of 1-forms {ω i }, e i , ω j = δ j i , and conversely; in particular ∂ µ , dx λ = δ λ µ . The exterior derivative decomposes as d = ω i e i . It is the same in the noncommutative case. The -dual can be obtained from {ω i } via a X -linear transformation that is the identity at zero order in ν [1], and the exterior derivative decomposes also as d = θ i e i . Using the -pairing we can associate to any 1-form ω the left X -linear map , ω : Ξ → X , and to any vector field X the right X -linear map X, : Ω → Ω . The maps i X := X, , i ω := , ω are the simplest twisted insertions (interior products) of a vector field in a form and of a 1-form in a multivector field, respectively. Using the exterior derivative and the twisted insertion, Lie bracket, and Lie derivative one can develop [54] a twisted Cartan calculus in complete analogy with the usual one (see also the thesis [55] for more details). As one can extend the commutative pairing to higher tensor powers setting for all X i ∈ Ξ, ω i ∈ Ω, so can one extend X, to the corresponding twisted tensor powers using the same formula (40). Properties (41), (44) are preserved. There is a -pairing , : Ω ⊗ Ξ → X with forms on the left and vector fields on the right. It is related to the previous -pairing via ω, X = R 1 X, R 2 ω for all ω ∈ Ω and X ∈ Ξ . It is left and right X -linear and satisfies ω h, X = ω, h X for all h ∈ X . As in the case of , there is an extension of , to higher twisted tensor powers.

Covariant derivative, torsion, curvature, metric
In [2, 1] a twisted covariant derivative (or, synonymously, twisted connection) ∇ F is defined as a collection of maps On functions the twisted covariant and Lie derivatives along X coincide, by (47). Eq. (51) amounts to the compatibility of the action of ∇ F X on 1-forms with the pairing of the latter with vector fields. ∇ F is left X -linear in the first argument, by (48); it is only C[[ν]]-linear in the second argument, by (49) and (50) with this holds in particular on vector fields T = Y . Actually, the knowledge of ∇ F just for (p, q) = (0, 0) (i.e. on functions) and (p, q) = (0, 1) (i.e. on vector fields), determines its unique extension to all the (p, q) ∈ N 2 0 : eq. (51) determines the action of ∇ F X on 1-forms, while (50) allows to extend ∇ F X recursively to all the T p,q 's, which consist of combinations of tensor products of 1-forms and vector fields.
The torsion T F and the curvature R F associated to a connection ∇ F are left X -linear maps for all X, Y, Z ∈ Ξ . They fulfill T F (X, h Y ) = T F (X h, Y ) (and similarly for the curvature), and the antisymmetry property Thus, one can regard torsion and curvature as elements of the following -tensor spaces acting on vector fields through the twisted pairing (40) applied to higher tensor powers, see (46). We omit the in the subscript of the elements (57) in order to distinguish them from the corresponding maps. In other words, for all X, Y, A metric is defined as a non-degenerate element g in the module If a twisted LC connection exists, it is unique by [2] Theorem 5. For equivariant metrics there is an existence and uniqueness theorem (c.f. [54] Lemma 3.12) of an equivariant twisted LC connection. If F = 1⊗1 (whereby becomes the ordinary product, and R = 1⊗1), the above formulae reduce to the notions and properties of ordinary connection, torsion, curvature, metric, etc. In particular we recover the characterization of a LC connection: In the commutative case the Ricci tensor is a contraction of the curvature tensor, Ric jk = R ijk i . The twisted Ricci map is defined as the following contraction of the curvature: where sum over i and (45) are understood. , is a contraction between forms on the left and vector fields on the right, see Section 2.2.1. Recall that it is defined by the pairing and has the X -linearity properties The twisted Ricci map is well defined because (62) is independent of the choice of the frame {e i } (and of the dual frame {θ i }), and because it is defined as a X -linear map: Evaluating the Ricci map on the dual metric yields the Ricci scalar: We now show how to construct nontrivial twisted deformations ∇ F of ∇. First, we need some preliminary result in ordinary differential geometry. It is easy to check that is a Lie subalgebra of Ξ; we shall name it the equivariance Lie algebra of ∇. It follows Proposition 2 Given a connection ∇ on M and the associated equivariance Lie algebra e, and satisfies the additional deformed Leibniz rule (with functions multiplying from the right) Of course, nontrivial deformations of this kind are possible only if e = {0}. We recall that Z ∈ Ξ is a Killing vector field of a (pseudo)Riemannian manifold (M, g) if or equivalently if g ∇ X Z, Y + g X, ∇ Y Z = 0 6 . The Killing vector fields close a Lie subalgebra k ⊂ Ξ; this is the Lie algebra of the group of isometries of (M, g) if M is complete.

Proposition 3
The Killing vector fields k ⊂ Ξ form a Lie subalgebra of the equivariance Lie algebra e of the Levi-Civita connection ∇ on a (pseudo)Riemannian manifold (M, g). For all ] the map g is also right X -linear in the second argument and related to the undeformed one g : and ∇ F X is the unique twisted Levi-Civita connection corresponding to g . Torsion and curvature of the twisted Levi-Civita connection remain undeformed as elements of the tensor spaces and the associated maps T F , R F are also right X -linear in the last argument. Eq. (60) boils down to The proofs of these propositions are in the appendix. The existence and uniqueness of the twisted Levi-Civita connection of Proposition 3 was proven in [2] Theorem 6 and Theorem 7. In NCG right function-linearity of the curvature in the last argument is in general not true, see e.g. [21,27]. In section 4 we will find nontrivial k for suitably symmetric quadrics in R 3 .
3 Twisted smooth submanifolds of R n

Differential geometry of manifolds embedded in R n
We develop some theoretical tools for the (n − k)-dimensional submanifolds M c ⊆ D f ⊆ R n defined by equations (1). Recall definitions (4), (10). We can identify Ξ M ⊂ Ξ with the Lie subalgebra of smooth vector fields tangent to M at all points, and Ξ t ⊂ Ξ with the Lie subalgebra of smooth vector fields tangent to all M c (c ∈ f (D f )) at all points, because .., k; as the Jacobian matrix J = (f a i ) has by assumption rank k, dim(Ξ t ) = n − k =: m. Henceforth Ω will stand for the X -bimodule of differential 1-forms on D f , i.e. the dual one of Ξ. We also define a X -subbimodule Ω ⊥ ⊂ Ω of 1-forms by . Ω ⊥ can be explicitly decomposed as Proof ω = ω a df a implies X, ω = X, df a ω a = X(f a )ω a = 0. Conversely, in any basis has rank m and fulfills f a i X i α = 0; decomposing ω = ω i dx i , Ξ t , ω = 0 amounts to X α , ω = X i α ω i = 0 for all α, and this linear system of m independent equations admits only solutions ω i = f a i ω a , ω a ∈ X , whence ω = ω a df a . This proves (79). For all X, W ∈ Ξ t , Y ∈ Ξ C , Z ∈ Ξ CC , a = 1, ..., k, h ∈ X , ω ∈ Ω ⊥ we find

Metric, Levi-Civita connection, intrinsic and extrinsic curvatures
We now discuss Ξ t , Ω ⊥ as addends in the decomposition of Ξ, Ω with respect to a metric. Consider a (non-degenerate) metric g ≡ g α ⊗ g α ∈ Ω ⊗ Ω on D f (actually, the following discussion is valid on any smooth manifold) and its dual is an isomorphism of X -bimodules with inverse given by ω → X ω = g −1α , ω g −1 α . In fact whenever ω = G(X), or equivalently X = G −1 (ω). Let us now introduce the X -subbimodules and let D f ⊆ D f be the open subset where the restriction is non-degenerate. If g is Riemannian, then D f = D f . For simplicity, henceforth we shall denote the restrictions of Ξ, Ξ t , Ξ ⊥ Ω, Ω ⊥ , Ω t to D f by the same symbols, and by k ⊂ Ξ t the Lie subalgebra of Killing vector fields of g that are also tangent to the submanifolds M c ⊂ D f .

Proposition 5
The Lie algebra Ξ of smooth vector fields and the X -bimodule Ω of 1-forms on D f split into the direct sums of X -subbimodules orthogonal with respect to the metric g and g −1 respectively. Ξ t is a Lie subalgebra of Ξ. Ω t is orthogonal to Ξ ⊥ with respect to the pairing: Ω t = {ω ∈ Ω | Ξ ⊥ , ω = 0}. Also the restrictions of g −1 to the tangent forms and of g to the tangent and normal vector fields are non-degenerate. The orthogonal projections pr ⊥ : Ξ → Ξ ⊥ , pr t : Ξ → Ξ t , pr ⊥ : Ω → Ω ⊥ , pr t : Ω → Ω t are uniquely extended as projections to the bimodules of multivector fields and higher rank forms through the rules pr ⊥ (ωω ) = pr ⊥ (ω)pr ⊥ (ω ), pr t (ωω ) = pr t (ω)pr t (ω ),...: Ξ t , Ξ ⊥ , Ω t , Ω ⊥ , their exterior powers and the projections pr ⊥ , pr t are U k-equivariant.
, Ω t = 0 and therefore also X ω , Ω = 0, whence by the non-degeneracy of the pairing, X ω = 0, and in turn ω = 0, namely g −1 Since g is non-degenerate, for all X ∈ Ξ t there is Y ∈ Ξ, and hence also Y t ∈ Ξ t , such that 0 = g(X, Y ) = g(X, Y t ): g t is non-degenerate. Similarly one proves that also g ⊥ is.
Remarks: i) The non-degeneracy of g −1 ⊥ (or, equivalently, of g −1 t ) is not only sufficient, but also necessary to ensure that hence ω belongs to Ω t as well. ii) Similarly, the non-degeneracy of g ⊥ (or, equivalently, of g t ) is necessary for Ξ ⊥ ∩ Ξ t = {0}. iii) While Ξ t is a Lie subalgebra of Ξ, in general Ξ ⊥ is not. iv) In general Ξ ⊥ , Ω t , and therefore also the orthogonal projections pr ⊥ , pr t , are not U Ξ t -equivariant; for this reason in section 3.2.1 we are able to deform (pseudo)Riemannian geometry only via twists based on k ⊂ Ξ t . v) We refer to elements of Ξ ⊥ , Ω ⊥ and Ω t as normal vector fields, normal 1-forms and tangent 1-forms.
As said, we identify Ξ t ⊂ Ξ with the Lie subalgebra of smooth vector fields tangent to all M c (c ∈ f (D f )) at all points, because X(f a ) = 0 implies X(f a c ) = 0; and Ξ M ⊂ Ξ defined in (4) with the Lie subalgebra of smooth vector fields tangent to M at all points. Similarly, we can identify Ω t with the subbimodule of Ω tangent to all M c (c ∈ f (D f )) at all points. We find Ωf a ⊂ Ω C . It fulfills Ξ, Ω CC ⊂ C. We can identify the X M -bimodule of 1-forms Ω M on M with the quotient Proposition 6 For all X ∈ Ξ C , ω ∈ Ω C , the tangent projections X t ∈ Ξ t , ω t ∈ Ω t belong to [X] ∈ Ξ M and [ω] ∈ Ω M respectively; similarly for multivector fields and higher rank forms.
Consequently, we can represent every element of Ξ M , Ω M , or more generally Ξ Mc , Ω Mc , resp. by an element of Ξ t , Ω t ; etc. In the appendix we prove Proposition 6, as well as the relations We call the restriction g t in (86) of the metric map g first fundamental form for the family of manifolds M c ⊂ D f , c ∈ f D f . It is X -linear in both arguments and further satisfies g t (X · h, Y ) = g t (X, h · Y ) for all X, Y ∈ Ξ t and h ∈ X (middle-linearity). Since g t is uniquely determined (via the pairing) by the tangent projectiong t = (pr t ⊗ pr t )(g) ∈ Ω t ⊗ Ω t of the metric g ∈ Ω ⊗ Ω, when there is no risk of confusion we will drop the tilde and with a slight abuse of notation denoteg t by g t . It is a symmetric element, i.e. τ (g t ) = g t . The first fundamental form (induced metric) on M is obtained by the further projection X → X M , which amounts to choosing the c = 0 manifold M out of the family. The same prescription will hold for the the Levi-Civita connection, curvature, etc., on M . Applying the decomposition of Ξ in tangent and normal vector fields to the restriction of the Levi-Civita connection we obtain the projected Levi-Civita connection and the second fundamental form for the family of manifolds M c : Proposition 7 The first fundamental form g t , the second fundamental form II and the projected Levi-Civita covariant derivative ∇ t are U k-equivariant maps.
Proof As compositions of U k-equivariant maps, g t , ∇ t and II are U k-equivariant.
By the Leibniz rule for ∇ and the X -linearity of pr t , pr t (hZ) = h pr t (Z) for all h ∈ X , Z ∈ Ξ, ∇ t is X -linear in the first argument, ∇ t,hX Y = h∇ t,X Y , and fulfills the Leibniz rule in the second argument, for all h ∈ X and X, Y ∈ Ξ t . Similarly, we find that II is X -linear in both arguments. By applying the further projection X → X

Decompositions in bases of Ω, Ξ; Euclidean, Minkowski metrics
In this section we explicitly determine the geometry (in particular, the decompositions (84) and the associated projections pr t , pr ⊥ ) in terms of bases of Ω, Ω ⊥ , Ω t and Ξ, Ξ ⊥ , Ξ t for a generic metric g, specializing to the Euclidean and Minkowski metrics at the end.
Let (x 1 , ..., x n ) be a n-ple of Cartesian coordinates; we lower and raise indices i, j, ... using the metric components g ij := g(∂ i , ∂ j ) and the dual ones Thus we can write the metric and its dual in the form implying, for all vector fields in terms of Cartesian coordinates) is symmetric and invertible, by (79), (83); we denote its inverse by K := E −1 . If the metric g is Riemannian, then E is also positive-definite on and, for all ω ∈ Ω, X ∈ Ξ, or, explicitly in terms of the decompositions ω (sum over repeated indices: h, i, j, ... run over 1, ..., n, while a, b, c, d, ... run over 1, ..., k).
The df a , N a ⊥ as well the E ab , K ab are k-invariant. On X ∈ Ξ, ω ∈ Ω the action of the projections pr ⊥ , pr t explicitly reads pr ⊥ (X) = X ⊥ , Proof We have already proved in Proposition 4 that B ⊥ is a basis of Ω ⊥ . As a consequence, ω ⊥ ∈ Ω ⊥ . From the definition we find g(X, N a ⊥ ) = K ab X i f b i = K ab X(f b ) = 0 for all X ∈ Ξ t and a = 1, ..., k, whence N a ⊥ ∈ Ξ ⊥ ; moreover, N a ⊥ (f b ) = K ac f ci ∂ i (f a ) = K ac E cb = δ ab , and N ⊥ is the basis of Ξ ⊥ dual to B ⊥ . As a consequence, X ⊥ ∈ Ξ ⊥ . g df a = 0 for all g ∈ Ξ t holds in particular for g ∈ k. By Proposition 7 g N a ⊥ ∈ Ξ ⊥ for all g ∈ k, and therefore g N a ⊥ = C a c (g)N c ⊥ with some coefficients C a c (g). Applying g to both sides of (99) and using the Ξ-equivariance of the pairing we thus find the k-invariance also of the N a ⊥ : is a straightforward computation; their kinvariance follows from that of df a and the U k-equivariance of g; in fact, ∀g ∈ U k The linear maps X → X ⊥ ∈ Ξ ⊥ , ω → ω ⊥ ∈ Ω ⊥ indeed realize the projection pr ⊥ , because hence also the linear maps X → X t := X−X ⊥ , ω → ω t := ω−ω ⊥ realize the projection pr t .
Remark. If g is Riemannian, setting H : are othonormal bases of Ξ ⊥ , Ω ⊥ , respectively and are dual to each other, i.e.
The k-invariance of θ a , U a ⊥ follows from that of df a , N a ⊥ and of E. In terms of the bases {U a ⊥ } k a=1 , {θ a } k a=1 the normal components of X ∈ Ξ, ω ∈ Ω read ω ⊥ = θ a g −1 (θ a , ω), Even if g is not Riemannian one can find in D f a k × k symmetric matrix H, such that , Ω ⊥ respectively that are othonormal up to suitable signs a = ±1 and dual to each other, in the sense where ζ ab = ζ ab := a δ ab (no sum over a). The normal components of X ∈ Ξ, ω ∈ Ω read If g is the Euclidean metric (g ij = δ ij ) the associated Levi-Civita connection on R n is We endow M ⊂ R n with the induced metric g t . Using X, Y, Z ∈ Ξ t as representatives of elements of Ξ M , the Levi-Civita connection on (M, g t ) is ∇ t,X Y := (∇ X Y ) t : (61), (74) hold with g, ∇, T, R replaced by where we have abbreviated f a ij := ∂ i ∂ j (f a ) ; thus, the second fundamental form II(X, Y ) := (∇ X Y ) ⊥ takes the explicit form Replacing this result and R = 0 in the Gauss equation (93), we find for the intrinsic curvature In fact, this condition guarantees that Z is Killing on (M c , g t ) for all c. The Killing vector fields close the Lie algebra k = h ∩ Ξ t of the group of isometries K of the M c 's; K is a subgroup of the group H of isometries of R n , i.e. of the Euclidean group (every element of H is a composition of a rotation, a translation and possibly an inversion of axis).
If g is the Minkowski metric [g ij = g ij = η ij = diag(1, ..., 1, −1)], the associated Levi-Civita connection on R n is again as in (104). Endowing M c ⊂ D f with the induced metric g t and using X, Y, Z ∈ Ξ t as representatives of elements of Ξ Mc , the Levi-Civita connection on (M c , g t ) is again ∇ t,X Y := (∇ X Y ) t : (61), (74) hold with g, ∇, T, R replaced by g t , ∇ t , T t , R t . In terms of components the condition for Z ∈ Ξ t to be a Killing vector field on (M c , g t ) remains (106).
Bases and complete sets of Ξ t , Ω t As seen, are respectively complete in Ω t , Ξ t , but are not bases, because of the linear dependence relations ϑ j f a j = 0, f aj W j = 0, a = 1, ..., k.
The above definition of B ⊥ does not involve any specific metric, as the definition (78) of Ω ⊥ itself. Similarly, as the definition (10) of Ξ t does not involve any metric, there should be some alternative complete set in Ξ t with the same feature. To determine it we start with the case k = 1, i.e. with a (n−1)-dimensional (hyper)surface M ⊂ D f determined by a single equation Rescaling S W by the factor f i f i we obtain another complete set: In fact, L ij annihilate f ; S L is complete because V j = f i L ij . This is the searched set, because its definition does not involve the metric. Clearly L ij = −L ji , so at most n(n−1)/2 of the L ij (e.g. those with i < j) are linearly independent over R. Obviously, both S V , S L are of rank n−1 over X ; they are respectively characterized by the dependence relations (here and below square brackets enclosing indices mean a complete antisymmetrization of the latter). As known, if M is not parallelizable there is no basis (i.e. complete set of just (n−1) elements) of Ξ t consisting of globally defined vector fields: redundancy is unavoidable. In the case of spheres f ≡ (x i x i − R 2 )/2 = 0 the n(n−1)/2 L ij := x i ∂ j − x j ∂ i (i < j) are the usual generators of rotations (angular momentum components), i.e. span so(n). The L ij are antihermitean under the * -structure (116), namely L * ij = −L ij . By an explicit computation we find that their Lie brackets are Now we consider the general k case. The globally defined vector fields are antihermitean, fulfill L i 1 i 2 ...i k+1 f a = 0 for all a = 1, ..., k, are completely antisymmetric with respect to (i 1 , i 2 , ..., i k+1 ), and make up a set S L complete (over X ) in Ξ t , independently of the metric. The L i 1 i 2 ...i k+1 with i 1 < i 2 < ... < i k+1 , or a subset thereof, is linearly independent over C. Even the latter may be linearly dependent over X , because f a [j L i 1 i 2 ...i k+1 ] = 0 for all a. We do not compute their Lie brackets here.

Differential calculus algebras
Henceforth we abbreviate ξ i := dx i . Let S = {e α } A α=1 be a set of vector fields, globally defined on D f that is complete in Ξ. The e α , ξ i fulfill relations of the type (with suitable t a l , C γ αβ ∈ X ). The first line contains possible linear dependence relations among the e α , like (111). If we choose S = {∂ 1 , ..., ∂ n } this is empty, while in the second line C γ αβ ≡ 0. Clearly the coefficients in the decomposition X = X α e α ∈ Ξ are defined up to shifts X α → X α + l h l t α l , with h l ∈ X . Consider the unital algebra Q • over C consisting of polynomials in ξ i , e α with (left or right) coefficients in X , modulo relations (114) and the ones It is easy to check that a different choice of S changes (114-115), but leads to an equivalent definition of Q • (one could choose also a different basis of 1-forms, but we will not consider this here). We shall name Q • differential calculus algebra on D f . The elements of Q • can be considered as differential-operator-valued inhomogenous forms. Relations (114-115) encode all the information about the differential calculus and allow to order the ξ i , e α in any prescribed way, with the coefficient functions at the left, center, or right -as one wishes. Q • admits X , Ω • , H as subalgebras; the enlarged Heisenberg algebra H is the component of form degree zero. While Q • , Ω • are graded by the form degree, Q • , H are filtered by the degree r in the e α ; r gives the order of an element of H seen as a differential operator on X . Note that within Q • also the action of a generic vector field X = X α e α on a function h can be expressed as a commutator: [X, h] = [X α e α , h] = X α [e α , h] = X(h). In the Q • framework Xh = hX + X(h) is the inhomogeneous first order differential operator sum of a first order part (the vector field hX) and a zero order part (the multiplication operator by X(h)); it must not be confused with the product of X by h from the right, which is equal to hX and in the previous sections has been denoted in the same way. In the Q • framework we denote the latter by X h (of course (X h)(h ) = X(h )h = hX(h ), X (hh ) = hh X remain valid). We endow Q • with the natural * -struture defined by If one chooses S so that a subset S t : a, b ∈ {1, ..., k}) invertible everywhere, then if α, β ≤ B the sum in (114) 2 is extended over γ ≤ B. The differential calculus algebra Q • Mc on M c is the X Mc -bimodule generated by the ξ 1 , ..., ξ n , e 1 , ..., e B modulo the relations (114-115) (with α, β ≤ B) and the ones 3.2 Twisted differential geometry of manifolds embedded in R n Using a twist F ∈ U Ξ t ⊗ U Ξ t [[ν]] and following the general twisting approach we deform the differential geometry on D f in a way compatible with the embeddings, i.e. so that it projects to the twist deformation of the differential geometry on the submanifolds M c , c ∈ R n . Equivalently, we deform the differential calculus algebra Q • on D f into an associated Q • in a way compatible with the embeddings, i.e. encoding through projections all deformations Unless explicitly stated, we still denote by X h = (R 1 h) (R 2 X) the vector field that is -product of the one X by the function h from the right, as done so far.
To state the twisted analog of Proposition 4 we first define a X -subbimodule Ω ⊥ ⊂ Ω : Proposition 9 Equipped with the -Lie bracket [ , ] Ξ t , Ξ C are -Lie subalgebras of Ξ , and Ξ CC is an ideal of Ξ C . Another -Lie algebra is thus . Ω ⊥ can be explicitly decomposed as Proof These are direct consequences of the following properties. By Proposition 4, for all  (14) and the relation g df a = ε(g)df a .
• X, ω = F 1 X, This means in particular that taking the quotient commutes with twisting. To build explicit examples of twist-deformed submanifolds we recall that several known types of Drinfel'd twists (as the ones mentioned in section 2.1.1) are based on finite-dimensional Lie algebras. When does the infinite-dimensional Ξ t admit a finite-dimensional Lie subalgebra g over R, so that we can choose F ∈ (U g⊗U g)[[ν]]? Given a set S of vector fields that is complete in Ξ t , the question is which combinations (with coefficients in X ) of them, if any, close a finite-dimensional Lie algebra g. An easy answer is available for the quadrics in R n , see section 4. If R n endowed with a metric admits a family M c of (pseudo)Riemannian submanifolds manifestly symmetric under a Lie group K (its group of isometries) 8 , then a nontrivial g exists and contains the (Killing) Lie algebra k of K (if M c is maximally symmetric then k is even complete -over Xin Ξ t ). In the next subsections we consider such a case and stick to deformations induced by a twist F based on k ⊂ Ξ t ; under these assumptions the deformation is compatible with the geometry. Ξ t , Ω ⊥ appear as addends in the decomposition of Ξ in tangent and orthogonal vector fields. In Section 3.2.2 we first give explicit results for a generic metric, then specialize the discussion to the Euclidean and Minkowski metric.

Twisted metric, Levi-Civita connection, intrinsic and extrinsic curvatures
As seen in section 3.1.1, endowing D f ⊆ R n with a (non-degenerate) metric g makes all the Again, let k ⊂ Ξ t the Lie subalgebra of Killing vector fields of g that are also tangent to the submanifolds M c ⊂ D f . The twisted version of Proposition 5 reads ] the -Lie algebra Ξ of smooth vector fields and the X -bimodule Ω of 1-forms on D f split into the direct sums of X -subbimodules orthogonal with respect to the twisted metrics g and g −1 respectively. Ξ t is a -Lie subalgebra of Ξ . Ω t , Ξ ⊥ are orthogonal with respect to the -pairing, Ω t = {ω ∈ Ω | Ξ ⊥ , ω = 0}. Also the restrictions of g −1 (resp. g) to the tangent and normal 1-forms (resp. vector fields) Similarly for -tensor powers of the former. The orthogonal projections pr ⊥ : Ξ → Ξ ⊥ , pr t : Ξ → Ξ t , pr ⊥ : Ω → Ω ⊥ , pr t : Ω → Ω t are uniquely extended as projections to the bimodules of multivector fields and higher rank forms through the rules pr ⊥ (ω ω ) = pr ⊥ (ω) pr ⊥ (ω ), pr t (ω ω ) = pr t (ω) pr t (ω ),...: pr ⊥ : Ω p → Ω p ⊥ , pr t : Ω p → Ω p t , pr ⊥ : pr ⊥ , pr t are the C[[ν]]-linear extensions of pr ⊥ , pr t . Ξ t , Ξ ⊥ , Ω t , Ω ⊥ , their -exterior powers and the projections pr ⊥ , pr t are U k F -equivariant.
Again we stress that, while Ξ t is a -Lie subalgebra of Ξ , in general Ξ ⊥ is not. Furthermore, as Ξ ⊥ , Ω t are not U Ξ F t -equivariant, also the orthogonal projections pr ⊥ , pr t are not. Proof By Proposition 9 Ξ t is a -Lie subalgebra of Ξ and a U Ξ F t -equivariant Xsubbimodule; in particular it is U k F -equivariant. Moreover, according to Proposition 3, as claimed. This also implies the equality Note that g −1 (ω, α) = g −1 (F 1 ω, F 2 α) for all ω, α ∈ Ω, since F is based on Killing vector fields. Now assume that X ∈ Ξ ⊥ (resp. X ∈ Ξ t ) fulfills g ⊥ (X, Ξ ⊥ ) = 0 (resp. g t (X, Ξ t ) = 0). Expanding X and g ⊥ (resp. g t ) in ν-powers and arguing as above, we find X = 0, whence the non-degeneracy of g ⊥ (resp. g t ). By employing (40) on Ω . Let X ∈ Ξ ⊥ , ω ∈ Ω t , ξ ∈ U k F . Then since g and the -pairing are equivariant under the action of U k F and Ξ t is a U k F -equivariant X -bimodule. This proves that also Ξ ⊥ , Ω t are U k-equivariant X -bimodules. To verify that pr ⊥ is U k F -equivariant let X = X t + X ⊥ ∈ Ξ be the decomposition (122) with X t ∈ Ξ t and X ⊥ ∈ Ξ ⊥ . Then ξ X = ξ X t +ξ X ⊥ and according to the for all ξ ∈ U k F . Similarly one argues with pr t on Ω and on the -exterior powers of Ξ , Ω . Finally, Ω t = {ω ∈ Ω | Ξ ⊥ , ω = 0} follows from its undeformed counterpart and the previous results.
As in the undeformed case, we identify Ξ t ⊂ Ξ with the -Lie subalgebra of smooth vector fields tangent to all M c (c ∈ f (D f )) at all points, because X(f a ) = 0 implies X(f a c ) = 0; and Ξ M ⊂ Ξ defined in (119) with the twisted Lie subalgebra of smooth vector fields tangent to M at all points. Similarly, we identify Ω t with the subbimodule of Ω tangent to all M c (c ∈ f (D f )) at all points. We find Ω t ⊂ Ω C : We can identify the X M -bimodule of 1-forms Ω M on M with the quotient Proposition 11 For all X ∈ Ξ C , ω ∈ Ω C the tangent projections X t := pr t (X) ∈ Ξ t , ω t := pr t (ω) ∈ Ω t respectively belong to [X] ∈ Ξ M and [ω] ∈ Ω M Consequently, we can represent every element of Ξ M (resp. Ω M ) by an element of Ξ t (resp. Ω t ). Similarly for multivector fields and higher rank forms.
Proof By Propositions 9, 10 the twist-deformed spaces can be identified with formal power series of the undeformed ones and the twisted projections are given by the [[ν]]-linear extensions of the undeformed ones. The claim follows as a corollary of Proposition 6.
Motivated from the classical situation we define the twisted first and second fundamental form on the family of submanifolds M c by as well as the twisted projected Levi-Civita connection on the family of submanifolds M c In the following proposition we clarify the relation of these objects to their classical counterparts. In particular, that twist deformation and projection to the submanifold commute.
Proposition 12 ∇ F t is a twisted covariant derivative on the family of submanifolds M c . The twisted first fundamental form g t is a metric on the family, with corresponding twisted Levi-Civita covariant derivative ∇ F t . They, as well as the second fundamental form, are U k Fequivariant. In terms of the undeformed objects we obtain and Proof As a composition of U k F -equivariant maps, g t , II F and ∇ F t also are. Eq. (130) follows from (75). Since ∇ F X = ∇ F 1 X F 2 for all X ∈ Ξ we find (131) and (132) (see also [2] eq. 129). Then it follows from Proposition 7 that g t is a (non-degenerate) metric on the M c 's with twisted Levi-Civita covariant derivative given by ∇ F t .
A generalization of Proposition 12 to braided commutative geometry is in [54] Proposition 4.4. The twisted second fundamental form (128) yields the twisted extrinsic curvature of M . The twisted intrinsic curvature R F t is related to the twisted curvature R F of ∇ F on R n by the following quantum analogue of the Gauss equation (see appendix 6.7 for the proof): Proposition 13 For all X, Y, Z, W ∈ Ξ t the following twisted Gauss equation holds: The twisted first and second fundamental forms, Levi-Civita connection, curvature tensor, Ricci tensor, Ricci scalar on M are finally obtained from the above objects by applying the further projection X → X M , which amounts to choosing the c = 0 manifold M out of the M c family. Of course, by a different choice of c one can do the same on any other M c .

Decompositions in bases of Ω , Ξ ; Euclidean, Minkowski metrics
In this section we explicitly determine the twisted geometry induced by a twist F ∈ U k⊗U k[[ν]] (in particular, the decompositions (122)) in terms of bases of Ω , Ω ⊥ , Ω t and Ξ , Ξ ⊥ , Ξ t for a generic metric g on R n , specializing to the Euclidean and Minkowski metric at the end, as done in section 3.1.2. By Proposition 10 Ξ t , Ω ⊥ , Ξ ⊥ , Ω t are U k F -equivariant, and the projections pr ⊥ , pr t are U k F -equivariant. Here is the twisted analogue of Proposition 8 and the remark following it: , {θ a } k a=1 are -dual, othonormal (possibly up to signs a = ±1) bases of Ξ ⊥ , Ω ⊥ respectively, in the sense On X ∈ Ξ , ω ∈ Ω the actions of the projections pr ⊥ , pr t explicitly read pr ⊥ (X) = X ⊥ , pr ⊥ (ω) = ω ⊥ , and pr t (X) = X t = X − X ⊥ , pr t (ω) = ω t = ω − ω ⊥ ; the normal components explicitly read in terms of the mentioned bases, twisted product and metric.
Proof All statements but the last one are straightforward consequences of the choice of the twist and of Propositions 5, 8. As pr ⊥ , pr t are just the C[[ν]]-linear extensions of pr ⊥ , pr t (see Proposition 10), then pr ⊥ (ω) = ω ⊥ , pr ⊥ (X) = X ⊥ , with the right-hand sides as defined in (97), and pr t (X) = X t = X − X ⊥ , pr t (ω) = ω t = ω − ω ⊥ . Eq. (137) holds because any twist -product boils down to an ordinary product if one of the two factors is U k-invariant, and similarly g −1 (ω, ω ) = g −1 (ω, ω ), and g (X, X ) = g(X, X ), if one of the arguments is U k-invariant, by eq. (23), (75) (14); the order of the factors and of the arguments of g , g −1 can be freely changed, for the same reason and the symmetry of metric.
An equivalent alternative to (137) is where ζ ab = a δ ab . By the -bilinearity of g the above equations imply in particular in terms of the left and right decompositions ω In the latter formulae one can decompose df a , N a ⊥ , θ a , U a ⊥ themselves in the same way, if one wishes. By the previous propositions, every complete set of Ξ t , e.g. Θ t , is also a complete set of Ξ t ; similarly, every complete set of Ω t , e.g. S V , or S L , is also a complete set of Ω t .

If the metric is Euclidean
replacing the right-hand sides in (139) makes the latter more explicit.

Twisted differential calculus algebras
The twist deformation of the differential calculus algebra Q • on R n introduced in section 3.1.3 gives the one Q • , with the same generators e α , ξ i and relations where C γ αβ ∈ X are defined by the decomposition [e α , e β ] ≡ F 1 e α , F 2 e β = C γ αβ e γ . Q • is a U Ξ F -equivariant X -bimodule. We endow Q • with the * F -structure. Note the change of notation: in the Q • framework X h = (R 1 h) (R 2 X) + X (h), hence X h has a different meaning with respect to the previous sections, where it stood just for the first term at the right-hand side, i.e. for the -product of the vector field X by the function h from the right; in the Q • framework, we denote the latter by X h := (R 1 h) (R 2 X), so that we can abbreviate If one chooses S so that a subset S t := {e α } B α=1 (B := A−k) is complete in Ξ t (e.g. it consists of the L i 1 i 2 ...i k+1 ), while e B+a := V a ⊥ , then, if α, β ≤ B, the sum in (141) 2 is extended over γ ≤ B. The twisted differential calculus algebra Q • Mc on M c is the X Mc -bimodule generated by the ξ 1 , ..., ξ n , e 1 , ..., e B , modulo the relations (141-142) with α, β ≤ B and the ones

Examples of twisted algebraic submanifolds of R 3
We can apply the whole machinery developed in the previous two sections to twist deform algebraic manifolds embedded in R n , provided we adopt X = Pol • (R n ), etc. everywhere. We can assume without loss of generality that the f a be irreducible polynomial functions 9 . Following subsection 3.2, it is interesting to ask for which algebraic submanifolds M c ⊂ R n the infinite-dimensional Lie algebra Ξ t admits a nontrivial finite-dimensional subalgebra g over R, so that we can build concrete examples of twisted M c by choosing a twist F ∈ (U g ⊗ U g)[[ν]] of a known type. As said, manifestly symmetric M c are of this type.
We can easily answer this question when k = 1 and the L ij themselves close a finitedimensional Lie algebra g over R. This means that in (112) f ij =const, hence f (x) is a quadratic polynomial, and M is either a quadric or the union of two hyperplanes (reducible case); moreover g is a Lie subalgebra of the affine Lie algebra aff(n) of R n . More explicitly, if with some real constants a µλ = a λµ (µ, λ = 0, 1, ..., n), f i = a ij x j +a i0 , f ij = a ij are constant, and (112) has already the desired form L ih act as linear transformations of the coordinates x k : By an Euclidean transformation (this is also an affine one) one can always make the x i canonical coordinates for the quadric, so that a ij = a i δ ij (no sum over i), b i := a 0i = 0 if a i = 0. In [35] the authors first classify g and derive some general results from the only assumptions X = Pol • (R n ) and g ⊂ aff(n); in particular, that the global description of differential geometry on R n , M c in terms of generators and relations extends to their twist deformations, in such a way to preserve the subspaces of the differential calculus algebras consisting of polynomials of any fixed degrees in the coordinates x i , differential dx i and vector fields chosen as generators. Then they analyze in detail the twisted quadrics embedded in R 3 .
Here we just present two families of the latter as examples of applications of the formalism developed in the previous sections. We analyze (referring for details to [35]) few twist deformations of the following classes of quadrics in R 3 : (a) elliptic cylinders; (c) elliptic cone; (b) 1-sheet and (d) 2-sheet hyperboloids. As usual, we identify two quadric surfaces if they can be translated into each other via an Euclidean transformation. By a suitable one we can make the equation f (x) = 0 take a canonical (i.e. simplest) form, which we use to identify the class. In Figure 2 we summarize the characterizing signs, rank, associated symmetry Lie algebra g, and type of twist deformation that we perform; an example in each class is plotted in Figure 1.
on M g the second term vanishes and the first is tangent to M g , as it must be; and similarly on M h . Having assumed the Jacobian everywhere of maximal rank M g , M h have empty intersection and can be analyzed separately. Otherwise L ij vanishes on M g ∩ M h = ∅ (the singular part of M ), so that on the latter a twist built using the L ij will reduce to the identity, and the -product to the pointwise product (see the conclusions). M c parametrized by c ≡ −a 00 ∈ R. This splits into a class of connected manifolds (the 1-sheet hyperboloids) and a class of disconnected ones (the 2-sheet hyperboloids and the cone, which has two nappes separated by the apex -a singular point); all are closed, except the cone, whose apex gives the E f of the family. In either case E f is an algebraic variety. We note that, since the L ij = f i ∂ j − f j ∂ i ∈ g involved in the twist vanish on E f , the deformation automatically disappears on it, and the twisted algebraic variety is well-defined as the undeformed. For other examples of submanifold algebras that are not algebras of functions on smooth manifolds we refer the reader to the recent paper [16]. We devote a subsection to each family and a proposition to each twist deformation; propositions are proved in [35], where twist deformations also of the other classes of quadrics are discussed in detail. Throughout this section the -product X h of a vector field X by a function h from the right is understood in the Q , Q Mc sense

(a) Family of elliptic cylinders in Euclidean R 3
Their equations in canonical form (with a 1 = 1, a 3 = a 0i = 0) are parametrized by c ≡ −a 00 > 0, a ≡ a 2 > 0 and read For every a > 0, {M c } c∈R + is a foliation of R 3 \ z, where z is the axis of equations x 1 = x 2 = 0. The vector fields Alternatively, as a complete set in Ξ t instead of {L 12 , L 13 , L 23 } we can use S t = {L 12 , ∂ 3 }, which is actually a basis of Ξ t ; the Lie algebra g so(2) × R generated by the latter is abelian.
In [35] the reader can find the relations characterizing these two deformations in detail.
In the case a 1 = a 2 = 1 (circular cylinder of radius R = √ 2c embedded in the Euclidean The Killing Lie algebra k is abelian and spanned (over R) by S t . ∇ X Y = 0 for all X, Y ∈ S , whereas the only non-zero ∇ X Y , with X, Y ∈ S are The second fundamental form is explicitly given by II(X, Y ) = (∇ X Y ) ⊥ = −XỸ R N ⊥ , for all X, Y ∈ Ξ t ; here we are using the decomposition Z =ZL + Z 3 ∂ 3 of the generic Z ∈ Ξ t . Thus II is diagonal in the basis S t , with diagonal elements (i.e. principal curvatures) κ 1 = 0, κ 2 = −1/R. Hence the Gauss (i.e. intrinsic) curvature K = κ 1 κ 2 vanishes; R t = 0 easily follows also from R = 0 using the Gauss theorem. The mean (i.e. extrinsic) curvature is

The deformation via the abelian twist
, L}, because ∂ 3 commutes with all such X, so that F 1 X ⊗ F 2 = X ⊗ 1, and the projections pr ⊥ , pr t , stay undeformed, as shown in Proposition 10. These results determine ∇ F X Y for all X, Y ∈ Ξ and ∇ F t,X Y = ∇ t,X Y for all X, Y ∈ Ξ t via the function left -linearity in X and the deformed Leibniz rule for Y . The twisted curvatures R F , R F t vanish. Furthermore, for all X, Y ∈ S t i.e. the principal curvatures κ 1 = 0, κ 2 = 1/R, Gauss and mean curvatures are undeformed.

(b-c-d) Family of hyperboloids and cone in Minkowski R 3
Their equations in canonical form (with a 1 = 1) are parametrized by a = a 2 > 0, b = −a 3 > 0, c = −a 00 (c > 0, c < 0 resp. for the 1-sheet and the 2-sheet hyperboloids, c = 0 for the cone) and read The Lie algebra g is spanned by showing that the corresponding symmetry Lie algebra is g so(2, 1). The commutation To compute the action of F on functions it is convenient to adopt as new coordinates the eigenvectors of H In Ref. [35] the reader can find, first the detailed actions of H, E, E on y i , ∂ i , η i , then the twisted coproducts and antipodes of H, E, E , the star products and commutation relations among the generators H, E, E , y i , ∂ i , η i arising from this twist.
Let us now focus on the case 1 = a This covers the circular cone, the circular hyperboloid of one and two sheets. We endow R 3 with the Minkowski metric g : We find the components of the curvature, Ricci tensors, the Ricci scalar (or Gauss curvature) on M c applying the Gauss theorem. All diverge as c → 0 (i.e. in the cone M 0 limit). M c is therefore anti-de Sitter space AdS 2 if c > 0, the union of two copies of de Sitter space dS 2 if c < 0. Under twist deformation the curvature (and Ricci) tensor on R 3 remain zero. By propositions 10, 3, 12 on M c the first and second fundamental forms, as well as the curvature and Ricci tensors, stay undeformed as elements of the corresponding tensor spaces: , ... Only the associated multilinear maps of twisted tensor products g t : Ξ t ⊗ Ξ t → X , ..., 'feel' the twist; they are related to the undeformed maps through formulae (75), (131) and [compare also to [2] Theorem 7 and eq. (6.138)], and similarly for R F t . Also the Ricci scalar (or Gauss curvature) R F t remains the undeformed one −1/c. Finally, one can elaborate [35] also abelian twist deformations for the elliptic cone, (153) with c = 0, enlarging the Lie algebra g by a generator D = x i ∂ i = y i∂ i of dilatations, which commutes with all L ij and is also tangent to the cone (only), as D(f ) = 2f : D ∈ Ξ M .

Outlook and final remarks
Considering a generic embedded submanifold M ⊂ R n that consists of the solutions x of a set of k equations f a (x) = 0 (a = 1, ..., k), where f : D f ⊂ R n → R k is a k-ple of smooth functions with Jacobian matrix of rank k, in this work we have explicitly built its noncommutative analogue in the framework of Drinfel'd (cocycle) twist [19] deformation of differential geometry [1,2]. This can be considered as a successful result, also in the broader framework of deformation quantization [6,41] ; only the pointwise product is deformed into a (in general noncommutative) one, the -product. In other words, taking the quotient and performing the deformation commute: X M = (X /C) = X /C . The key point has been to perform the deformation using a Drinfeld twist F based on the Lie subalgebra Ξ t (10) of vector fields on D f that are tangent to every manifold M c of the family of level sets of f (the latter is parametrized by c ∈ f (D f ) ⊂ R k ; M c consists of the solutions of f a (x) − c a = 0, a = 1, ..., k), rather than on the Lie algebra Ξ M of vector fields tangent to M only; this has given for free the deformation of the whole family by the same twist. Every vector field in the -Lie algebra Ξ M can be represented by an element of the -Lie algebra Ξ t , as it occurs in the undeformed case. The whole twisted Cartan calculus is automatically equivariant under the non-cocommutative Hopf algebra U Ξ F t ; the latter may be interpreted as the quantum group of (small) diffeomorphisms of the deformed submanifolds. The dimensions of Ξ , Ξ t as X bimodules, as well as of their duals Ω , Ω t , remain undeformed, contrary to what happens to the quantum group bicovariant or equivariant differential calculi mentioned in the introduction. This is because we consider 2-cocycles twists, but could change with more general twists leading to quasitriangular Hopf algebras or quasi-Hopf algebras, or twists in the category of bialgebroids. We have also shown that, when R n is endowed with a connection ∇, taking the tangent projection (from R n to M ) of ∇ and the associated torsion, curvature, commutes with performing the deformation, provided F is based on the equivariance Lie subalgebra e ⊂ Ξ t [see (67)]. When R n is endowed with a metric g the same holds for g itself, the associated Levi-Civita connection, the intrinsic and extrinsic curvatures (while the torsions remain zero), only if F is based on the Lie algebra k ⊂ Ξ t of Killing vector fields of the metric.
All our results are global, in that we have determined global (i.e., defined on all of M ) bases -or complete sets -of all the relevant X -and X M -bimodules from their undeformed counterparts: C is spanned by the globally defined functions f a , Ξ t by some complete set {e α } of globally defined vector fields [e.g. (113)]; these fulfill some linear dependence relations], the X -bimodule Ξ ⊥ ⊂ Ξ of twisted vector fields normal to the M c 's (with respect to the metric g) is spanned by the globally defined vector fields (96), and similarly the dual ones Ω t , Ω ⊥ of 1-forms, their tensor or wedge powers,.... This means that both in the undeformed and deformed context these bimodules/algebras can be formulated in terms of (the mentioned) generators and polynomial relations, with elements in X , X as coefficients.
In the polynomial setting, if the polynomial functions f a (x) fulfill suitable irreducibility conditions, then also X , X , X M , X M can be defined in terms of generators x i (the Cartesian coordinates) and polynomial relations [35]. The procedure can be potentially applied to a large number of algebraic manifolds, starting from algebraic hypersurfaces (k = 1), in particular quadrics; one can use the examples of cocycle twists available in the literature (tipically based on finite-dimensional Lie algebras g) to build concrete deformations of these submanifolds. In [35] the authors discuss in detail deformations of all families of quadric surfaces embedded in R 3 that are induced by unitary twists of the abelian [51] or Jordanian [49,50] type, except the ellipsoids. Here (section 4) we have only presented the results for the elliptic (in particular, circular) cylinders, hyperboloids and cone. Endowing R 3 with the Euclidean (resp. Minkowski) metric we have found twisted circular (i.e. maximally symmetric) cylinders, hyperboloids and cone M c that are (pseudo)Riemannian and equivariant under a non-trivial Hopf algebra U k F ("quantum group of isometries"); the twisted Levi-Civita connection on all M c equals the projection of the twisted Levi-Civita connection on R 3 (the exterior derivative), while the twisted intrinsic curvature can be expressed in terms of the twisted second fundamental form (or extrinsic curvature) via the twisted Gauss theorem; the twisted curvatures are the same constants as their undeformed counterparts. The twisted hyperboloids with c < 0 (resp. c > 0) can be thus considered as twisted de Sitter spaces dS 2 (resp. anti-de Sitter spaces AdS 2 ).
We recall that the higher-dimensional generalizations of the latter manifolds play a prominent role in present cosmology and theoretical physics as maximally symmetric cosmological solutions to the Einstein field equations of general relativity with a nonzero cosmological constant Λ; in particular, de Sitter spacetime (Λ > 0) can describe a universe with accelerating expansion rate (see e.g. [17]), while anti-de Sitter spacetimes (Λ < 0) are at the base of the socalled Ads/CFT correspondence [46]. Interpreting Minkowski R 2+1 as a relativistic momentum, rather than position, space (x 1 , x 2 playing the role of components of the momentum, x 3 of energy), the equations (153) as dispersion relations for relativistic particles, and performing the deformations, we should regard the x 3 > 0 component of the twisted 2-sheet hyperboloid (c < 0) as the twisted mass shell of a particle of mass |2c|; similarly, the x 3 > 0 nappe of the cone c = 0 would do for a massless particle, while c > 0 would do for a tachyon.
Generalizing the framework to submanifolds of C n looks straightforward and should make things even simpler, as we drop * -structures and the related constraints on the twist. For instance, there are no abelian twist deformations of the ellipsoids in R 3 , because the corresponding g so (3)  Finally, in [33,34,36] an alternative approach to introduce noncommutative (fuzzy) embedded submanifolds S in R n was proposed and applied to the spheres; it is obtained projecting the algebra of observables of a quantum particle in R n , in a confining potential with a very sharp minimum on S, to the Hilbert subspace with energy below a certain cutoff.

Proof of Proposition 1
As the inclusion C ⊃ k a=1 X f a is trivial, we need to prove the converse one. For allx ∈ M we can find a local smooth change of coordinates φ : x ∈ Vx → z ∈ Uz of the form φ(x) = (f, y) ≡ (f 1 , ..., f k , y 1 , ..., y n−k ), wherez ≡ φ(x) = (0 k ,ȳ) (0 k stands for the row with k zeroes), Uz ⊂ R n is an open ball with centerz, and Vx = φ −1 (Uz) ⊂ R n ; one can choose the extra coordinates y h e.g. as a subset of the x j themselves 10 . For all h ∈ X the function defined on Uz byĥ(z) = h(x) is smooth as well. In terms of the new coordinates the points of Vx ∩ M belong to the hyperplane z 1 = ... = z k = 0. For all z = (c, y) ∈ Uz we denote as z := (0 k , y) its projection on this hyperplane; the segment zz is contained in the ball Uz. Applying Hadamard's lemma to the dependence ofĥ(z) on the first k coordinates [considering y as parameters] we findĥ(z) =ĥ(z ) + k a=1 c aĥa (z) in Uz, with smoothĥ a ; more explicitly, . This is the desired decomposition, but only locally. To make it global, consider the open cover of D f where M a is the closure of the hypersurface M a , which is the level set of f a (M = k a=1 M a ). Since D f is paracompact (as so is the metric space R n ), there is a smooth partition of unity subordinated to O, i.e. there exist: a function ρ a with support contained in V a , for all a ∈ {1, ..., k}, and a function ρx ∈ X with support contained in Vx, for allx ∈ M , such that for all x ∈ D f x∈M ρx(x) + k a=1 ρ a (x) = 1, with only a finite number of non-zero terms in the sum. The functions belong to X and fulfill x +h a ∈ X are the coefficients needed for (3) to hold. In fact, for all The Jacobian matrix of l = (l 1 , ..., l k ) is the k × (n+k)-matrix (J| − I k ), where J = ∂f /∂x has rank k. M consists of the points x such that (x, 0 k ) solves (159). Fixed ax ∈ M , we can always permute the coordinates so that the k × k-matrix A := ∂f a /∂x b k a,b=1 is invertible inx. By the implicit function theorem there exists an open ball Uz ⊂ R n centered atz := (0 k ,x k+1 , ...,x n ) and smooth functions x a (z) of z := (c 1 , ..., c k , x k+1 , ..., x n ) ∈ Uz such that x a (z) =x a , and l x 1 (z), ..., x k (z), x k+1 , ..., x n , c 1 , ..., c k = 0; thus we can set y 1 = x k+1 , ..., y n−k = x n .

More on twists
We write in a compact notation the inverse of (15) and its consequences obtained applying ∆ on the first, second, third tensor factor and recalling that ∆ is cocommutative; the bracket encloses tensor factors obtained from one by application of ∆. To denote the decomposition of F (12)3 we have used a Sweedler-type notation F (12)3 ≡ (∆ ⊗ id )(F) = F 1(1) ⊗F 1(2) ⊗F 2 , and similarly for F 1 (23) , F (12)3 .... Several proofs are based on these formulae.
Since g is non-degenerate it follows that ξ ∇ X Y = ∇ ξ (1) X (ξ (2) Y ), i.e. ξ is an element of the equivariance Lie algebra of ∇. Thus we have shown the inclusion k ⊂ g. If F ∈ U k ⊗ U k[[ν]], then F 2 g = ε(F 2 )g, and using (14), (27) we immediately find (75). In fact where in the last two equations the pairing is extended to double tensor products, see (46). This reduces to the undeformed g(X,Y ) if F = 1⊗1. The twisted metric g is right X -linear, since for all f ∈ X . Next we prove that ∇ F is torsion-free with respect to the twisted torsion and metric compatible with respect to the twisted metric. The first property holds since while the second one holds because for all X, Y, Z ∈ Ξ, which is equivalent to ∇ F X g = 0. The statements about the twisted curvature and torsion are proven in [2] Theorem 7, while the uniqueness of ∇ F is given by [2] Theorem 5. We prove that the twisted torsion and curvature are right -linear in the last argument if F is based on Killing vector fields. Let X, Y, Z ∈ Ξ and h ∈ X . Then hT F (X, Y ) h proves right X -linearity of the twisted torsion. Finally, where in the last equation the eighth term cancels with the fourth and seventh term, the second and sixth cancel each other, and so do the third and fifth terms.
6.6 Proof of Proposition 6 and eq. (89) Then X(f b ) = X a N a ⊥ (f b ) = X a K ac (f ci f b i ) = X b must belong to C for all b = 1, .., k, i.e. must be of the form X a = f b X a b , for some X a b ∈ X . Hence X ⊥ = f b (X a b N a ⊥ ) belongs to Ξ CC , and X t ∈ [X]. Decompose ω = ω t + ω ⊥ . One can find an atlas of D f , with a pair {e i }, {θ i } of dual frames in each chart, such that {e α } n−k α=1 is a basis of Ξ t and {θ α } n−k α=1 is a basis of Ω t . Then ω t = ω α θ α , and for all X = X α e α ∈ Ξ t it is X, ω = X, ω t = X α ω α ; by Theorem 1, this belongs to C for all (X α ) if and only if ω α = f a ω a α , for some ω a α ∈ X . Hence ω ⊥ = f a ω a α θ α belongs to Ω CC , and ω t ∈ [ω]. In Proposition 4 we have shown that Ω ⊥ ⊆ Ω ⊥ := {ω ∈ Ω | Ξ t , ω = 0}. Conversely, for any ω ∈ Ω ⊥ we have 0 = X, ω = X, ω t for all X ∈ Ξ t , whence ω t = 0 and ω = ω ⊥ ∈ Ω ⊥ . This proves the first equality in (89). To prove the last equality, decompose X = X t + X ⊥ ; this belongs to Ξ C if and only if X ⊥ is of the form X ⊥ = f b X ba N a ⊥ , whence X, ω = X t , ω + f b X ba N a ⊥ , ω belongs to C iff X t , ω does, for all X t ∈ Ξ t .