Twisted Quadrics and Algebraic Submanifolds in $\mathbb{R}^n$

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold $M$ of $\mathbb{R}^n$, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of $\mathbb{R}^n$, arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883-1911, 2006), whereby the commutative pointwise product is replaced by the $\star$-product determined by a Drinfel'd twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds $M_c$ that are level sets of the $f^a(x)$, where $f^a(x)=0$ are the polynomial equations solved by the points of $M$, employing twists based on the Lie algebra $\Xi_t$ of vector fields that are tangent to all the $M_c$. The twisted Cartan calculus is automatically equivariant under twisted $\Xi_t$. If we endow $\mathbb{R}^n$ with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted $M$ is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and $\star$-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $\mathbb{R}^3$ except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb{R}^3$ and twisted hyperboloids embedded in twisted Minkowski $\mathbb{R}^3$ [the latter are twisted (anti-)de Sitter spaces $dS_2$, $AdS_2$].

(see e.g. [15,12]). Surprisingly, the question whether, and to what extent, a notion of a submanifold is possible in NCG has received little systematic attention (rather isolated exceptions are e.g. Ref. [45,54,17,48]). On several noncommutative (NC) spaces one can make sense of special classes of NC submanifolds, but some aspects of the latter may depart from their commutative counterparts. For instance, from the SO q (n)-equivariant noncommutative algebra "of functions on the quantum Euclidean space R n q ", which is generated by n non-commuting coordinates x i , one can obtain the one A on the quantum Euclidean sphere S n−1 q by imposing that the [central and SO q (n)-invariant] "square distance from the origin" r 2 = x i x i be 1. But the SO q (n)-equivariant differential calculus on A (i.e. the corresponding A-bimodule Ω of 1-forms) remains of dimension n instead of n−1; the 1-form dr 2 cannot be set to zero, and actually the graded commutator 1 q 2 −1 r −2 dr 2 , · acts as the exterior derivative [28,53,26,11].
In [32] the above question is systematically addressed within the framework of deformation quantization [6], in the particular approach based on Drinfel'd twisting [21] of Hopf algebras; a general procedure to construct noncommutative generalizations of smooth submanifolds M ⊂ R n , of the Cartan calculus, and of (pseudo)Riemannian geometry on M is proposed. In the present work we proceed studying more in detail algebraic submanifolds M ⊂ R n , in particular quadrics, using tools of algebraic geometry. Considering C n instead of R n seems viable, too.
Assume that the algebraic submanifold M ⊂ R n consists of solutions x of the equations f a (x) = 0, a = 1, 2, ..., k < n, where f ≡ (f 1 , ..., f k ) : R n → R k are polynomial functions fulfilling the irreducibility conditions listed in Theorem 1; in particular, the Jacobian matrix J = ∂f /∂x is of rank k on some non-empty open subset D f ⊂ R n , and M more precisely consists of the points of D f fulfilling (1). One easily shows that E f := R n \ D f is empty or of zero measure 1 . 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 k-parameter family of embedded manifolds M c (M 0 = M ) of dimension n−k that are level sets of f . Embedded algebraic submanifolds N ⊂ M can be obtained by adding more polynomial equations of the same type to (1). Let X be the * -algebra (over C) of polynomial functions P : R n → C, restricted to D f . The * -algebra X M of complex-valued polynomial functions on M can be expressed as the quotient of X over the ideal C ⊂ X of polynomial functions vanishing on M : In appendix A, after recalling some basic notions and notation in algebraic geometry, we prove Theorem 1 Assume that J is of rank k on a non-empty open subset D f ⊂ R n , so that the system (1) defines an algebraic submanifold M ⊂ D f of dimension n − k. In addition, assume that M is irreducible in C n ; this is the case e.g. if there exists a k-dimensional affine subspace π ⊂ R n meeting M in s := k a=1 deg f a points. Then C is the complexification of the ideal generated by the f a in R[x 1 , ..., x n ], i.e. for all h ∈ C there exist h a ∈ X such that (In the smooth context, i.e. with f a , h, h a ∈ C ∞ (D f ), (3) holds if J is of rank k on D f [32].) X N is the quotient of X M over the ideal generated by further equations of type (1), or equivalently of X over the ideal generated by all such equations. Identifying vector fields with derivations (first order differential operators), we denote as Ξ := {X = X i ∂ i | X i ∈ X } the Lie algebra of polynomial vector fields X on D f (here and below we abbreviate ∂ i ≡ ∂/∂x i ) and Ξ C = {X ∈ Ξ | X(f a ) ∈ C for all a ∈ {1, . . . , k}}, The former is a Lie * -subalgebra of Ξ, while the latter is a Lie * -ideal; both are X - *subbimodules. By Theorem 1 the latter decomposes as Ξ CC = k a=1 f a Ξ. We identify the Lie algebra Ξ M of vector fields tangent to M with that of derivations of X M , namely with A general framework for deforming X into a family -depending on a formal parameter νof noncommutative algebras X over C [[ν]] (the ring of formal power series in ν with coefficients in C) is Deformation Quantization [6,40]: 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 , and fulfilling itself X ]. As a result, taking the quotient would commute with deforming the product: (X /C) = X /C . As argued in [32], these conditions are fulfilled if 2 , for all α ∈ X , a = 1, .., k, (this implies that the f a are central in X , again). The quotient (7) also appears in the context of deformation quantization of Marsden-Weinstein reduction [10,37]. A more algebraic approach to deformation quantization of reduced spaces is given in the recent article [18]. In [21] Drinfel'd 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) (here ⊗ = ⊗ C[[ν]] , and tensor products are meant completed in the ν-adic topology); F acts on the tensor product of any two U g-modules or module algebras, in particular algebras of functions on any smooth manifolds G acts on, including some symplectic manifolds 3 [3]. Given a generic smooth manifold M , the authors of [1] pick up g ≡ Ξ M , the Lie algebra of smooth vector fields on M (and 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 corresponding deformed product reads where is the inverse of the twist. In the sequel we will use Sweedler notation with suppressed summation symbols and abbreviate F = F 1 ⊗ F 2 , F = F 1 ⊗ F 2 ; in the presence of several copies of F we distinguish the summations by writing F 1 ⊗ F 2 , F 1 ⊗ F 2 , etc. Actually Ref. [1] twists not only U Ξ M , X M into new Hopf algebra U Ξ F M and U Ξ F M -equivariant 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 and in [32], as in [45], we take the algebraic characterization (2), (5) as the starting point for defining submanifolds in NCG, but use a twist-deformed differential calculus on it. Our twist is based on the Lie subalgebra (and X -bimodule) g ≡ Ξ t ⊂ Ξ defined by which consists of vector fields tangent to all submanifolds M c (because they fulfill X(f a c ) = 0 for all c ∈ R k ) at all points. As in [32], we note that, applying this deformation procedure to the previously defined X with a twist F ∈ U Ξ t ⊗ U Ξ t [[ν]], we satisfy (8) and therefore obtain a deformation X of X such that for all c ∈ f (D f ) X In other words, we obtain a noncommutative deformation, in the sense of deformation quantization and in the form of quotients as in (2), (5), of the k-parameter family of embedded algebraic manifolds M c ⊂ R n . 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 ; hence we can work with the latter. X , Ξ , ... are U Ξ F -equivariant, while X Mc , Ξ M , Ξ t , ... are U Ξ F t -equivariant. If F is unitary or real, then U Ξ F and X , Ξ , ... admit * -structures (involutions) making them a Hopf * -algebra and U Ξ F -equivariant (Lie) * -algebras respectively; thereby U Ξ F t is a Hopf * -subalgebra and X Mc , Ξ t , ... are U Ξ F t -equivariant (Lie) * -subalgebras. In passing, we recall that sometimes, if a Poisson manifold M is symmetric under a solvable Lie group G like R d , the Heisenberg or the "ax + b" group, one can construct even a strict (i.e. non-formal) deformation quantization [52] of C ∞ (M ) such that the -product remains invariant under G itself (or a cocommutative Hopf algebra), see e.g. [52,7].
In section 3 we apply this procedure to algebraic submanifolds M ⊂ R n . For simplicity we stick to M of codimension 1, and we assume that there is 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 ; the level sets of f (x) of degree 1 (hyperplanes) or 2 (quadrics) are of this type. Choosing a twist F ∈ U g⊗U g[[ν]] we find that the algebra X 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. This implies in particular that the polynomial relations x i x j − x j x i = 0 (whence the commutativity of X ), as well as the ones (1) defining the ideal C, can be expressed as -polynomial relations of the same degree, so that X , X M = X /C can be defined globally in terms of generators and polynomial relations, and moreover the subspaces X q , X M q of X , X M = X /C consisting of polynomials of any degree q in x i coincide as C[[ν]]-modules with their deformed counterparts X q , X M q ; in particular their dimensions (hence the Hilbert-Poincaré series of both X and X M ) remain the same under deformation -an important (and often overlooked) property that guarantees the smoothness of the deformation. The same occurs with the X -bimodules and algebras Ω • of differential forms, that of differential operators, etc. We convey all these informations into what we name the differential calculus algebras Q • , Q • Mc on R n , M respectively (generated by the Cartesian coordinates, their differentials, and a basis of vector fields, subject to appropriate relations; they are graded by the form degree and filtered by both the degrees in the x i and in the vector fields), and their deformations Q • , Q • Mc (see sections 3.1, 3.2). In section 4 we discuss in detail deformations, induced by unitary twistsof abelian [51] or Jordanian [49] type, of all families of quadric surfaces embedded in R 3 , except ellipsoids. The deformation of each element of every class is interesting by itself, as a novel example of a NC manifold. Endowing R 3 with the Euclidean (resp. Minkowski) metric gives the circular cylinders (resp. hyperboloids and cone) a Lie algebra k ⊂ Ξ t 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"); the twisted Levi-Civita connection on R 3 (the exterior derivative) projects to the twisted Levi-Civita connection on M c , while the twisted curvature can be expressed in terms of the twisted second fundamental form through a twisted Gauss theorem. Actually, the metric, Levi-Civita connection, intrinsic and extrinsic curvatures of any circular cylinder or hyperboloid, 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 appendices A,B we recall basic notions in algebraic geometry and prove most theorems. We recall that (anti-)de Sitter spaces, which can be represented as solutions of 2f c (x) ≡ (x 1 ) 2 +...+(x n−1 ) 2 −(x n ) 2 −2c = 0 in Minkowski R n , are maximally symmetric cosmological solutions to the Einstein equations of general relativity with a nonzero cosmological constant Λ in spacetime dimension n−1, and play a prominent role in present cosmology and theoretical physics (see e.g. [19,44]). Interpreting x in Minkowski R n as relativistic n-momentum, rather than position in spacetime, then the same equation represents the dispersion relation of a relativistic particle of square mass 2c. In either case it would be interesting to study the physical consequences of twist deformations. On the mathematical side, directions for further investigations include: submanifolds of C n (rather than R n ), just dropping * -structures and the related constraints on the twist; twist deformations of the (zero-measure) algebraic set E f .
Finally, we mention that in [30,31,50] an alternative approach to introduce NC (more precisely, fuzzy) submanifolds S ⊂ R n has been proposed and applied to spheres, projecting the algebra of observables of a quantum particle in R n , subject to a confining potential with a very sharp minimum on S, to the Hilbert subspace with energy below a certain cutoff.
Everywhere we consider vector spaces V over the field K ∈ {R, C}; we denote by V [[ν]] the K[[ν]]-module of formal power series in ν with coefficients in K. We shall denote by the same symbol a K-linear map φ :

Hopf algebras and their representations
Hopf algebras.
It is cocommutative, i.e. τ • ∆ = ∆, where τ is the flip, τ (a ⊗ b) = b ⊗ a. If there is a * -involution * : g → g on g such that [g, h] * = [h * , g * ] for all g, h ∈ g, the UEA U g becomes a Hopf * -algebra with respect to the extension * : U g → U g.
Replacing everywhere in the above definition K by the commutative ring K[[ν]] one obtains the definition of a Hopf ( * -)algebra over K [[ν]]. For any Hopf ( * -)algebra over K the K[[ν]]linear extension (with completed tensor product in the ν-adic topology) is trivially a Hopf ( * -)algebra over K [[ν]]. Other ones can be obtained by twisting (see below).
Hopf algebra modules and module algebras. Given an associative unital algebra A over K, a K-vector space M is said to be a left A-module if it is endowed with a K-linear map : A⊗M → M such that a (b s) = (a · b) s and 1 s = s for all a, b ∈ A and s ∈ M. Similarly right A-modules are defined. An A-bimodule is a left and a right A-module with commuting module actions. A K-linear map φ : M → M between left A-modules is said to be A-equivariant if φ intertwines the A-module actions, i.e. if φ(a s) = a φ(s) for all a ∈ A and s ∈ M. For a Hopf * -algebra H, a left H-module M is said to be a left H- * -module if there is a * -involution * : M → M on M such that (a s) * = S(a) * s * for all a ∈ H and s ∈ M.
Similarly, right A- * -modules and A- * -bimodules are defined. An element s ∈ M of a left H-module is said to be H-invariant if a s = (a)s for all a ∈ H. An associative unital ( * -)algebra A is said to be a left H-( * -)module algebra if A is a left H-( * -)module such that for all ξ ∈ H and a, b ∈ A. More generally, an A-( * -)bimodule M for a left H-module ( * -)algebra A is said to be an  (18) and the normalization property ( ⊗id)(F) = 1 = (id⊗ )(F). Every twist is invertible as a formal power series. We denote the inverse twist by F and suppress summation symbols, employing the leg notation: F = F 1 ⊗F 2 , F = F 1 ⊗F 2 , and F 1 ⊗F 2 ⊗F 3 for the expression at both sides of (18). In the presence of several copies of F we write F = F 1 ⊗F 2 for the second copy etc. to distinguish the summations. To every twist we assign an element β : It is invertible with inverse given by where the twisted coproduct and antipode are defined by for all ξ ∈ H 4 . Again, we shall use Sweedler's notation with suppressed summation symbols for the coproduct ∆ F and its (n−1)-fold iteration where R := As the representation theory of a Hopf algebra H is monoidal, the K-tensor product M⊗M of two left H-modules is also a left H-module, via the H action ξ (s⊗s ) = ξ (1) s⊗ξ (2) s . The -tensor product is the corresponding monoidal structure on the representation theory of H F , since  [24,29]). In other words, D(ξ ξ ) = D(ξ)D(ξ ), and ∆ , S , R are related to ∆ F , S F , R by the relations One can think of D also as a change of generators within If H is a Hopf * -algebra, and the twist is either real [namely, if F * ⊗ * = (S ⊗S)(F 21 )] or unitary (namely, if F * ⊗ * = F), then one can make both H F and H into Hopf * -algebras in such a way that twisting transforms the H * -modules and module * -algebras into H F and H * -modules and module * -algebras, respectively. In fact, if F is real then also β * = β, while Actually D is an isomorphism of the triangular Hopf * -algebra (H , * ) onto the one (H F , * F ), see [1,43] for more information. If F is a unitary twist, then also R is, β * = S β −1 , and H F endowed with the undeformed * -involution is a Hopf * -algebra; moreover, A , M are respectively a left H F -module * -algebra and an H F -equivariant A - * -bimodule when endowed with the twisted * -involutions where a ∈ i.) For a finite number n ∈ N of pairwise commuting elements e 1 , . . . , e n , f 1 , . . . , f n ∈ g we set P : is a Drinfel'd twist on U g ( [51]); it is said of abelian (or Reshetikhin) type. It is unitary if P * ⊗ * = P ; this is e.g. the case if the e i , f i are anti-Hermitian or Hermitian. The twist F = exp(iνP ) 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 (29).
ii.) Let H, E ∈ g be elements of a Lie algebra such that [H, E] = 2E. Then defines a Jordanian Drinfel'd twist [49]. If H and E are anti-Hermitian, F is unitary.

Twisted differential geometry
Here we recall some results obtained in [1,2]. We apply the notions overviewed in the previous section choosing as Hopf * -algebra H = U Ξ, where Ξ := Γ ∞ (T M ) denotes the Lie * -algebra of smooth vector fields on a smooth manifold M , as a left H-module * -algebra the * -algebra X = C ∞ (M ) of smooth K-valued functions on M , as H-equivariant symmetric X - * -bimodules Ξ itself, the space Ω = Γ ∞ (T * M ) of differential 1-forms on M , as well as their tensor (or wedge) powers. The Hopf * -algebra action on X , Ξ and Ω is given by the extension of the Lie derivative: for X, Y ∈ Ξ, f ∈ X and ω ∈ Ω we have and we set L XY = L X L Y , L 1 = id. Henceforth we denote such an extension by .

Twisted tensor fields
The tensor algebra T := p,r∈N 0 T p,r on M is defined as the direct sum of the K-modules for p, r ≥ 0, p + r > 0, where we set T 0,0 := X . Here and below ⊗ stands for ⊗ X (rather than ⊗ K ), namely T ⊗ f T = T f ⊗ T for all f ∈ X . Every T p,r is an H-equivariant X - * -bimodule with respect to the module actions for all ξ ∈ H and h, k ∈ X . This induces the structure of an H-equivariant X - * -bimodule on T . In particular, for all T, T ∈ T , ξ ∈ H and h, k ∈ X the relations hold. Let T ∈ T p,r . On a local chart (U, x) of M there are unique functions T λ 1 ,...,λr µ 1 ,...,µp ∈ C ∞ (U ) such that T = T λ 1 ,...,λr µ 1 ,...,µp dx µ 1 ⊗. . .⊗dx µp ⊗∂ λ 1 ⊗. . .⊗∂ λr , where {∂ i } is the dual frame of vector fields on U corresponding to {x i }, i.e. ∂ i , dx j = δ j i and we sum over repeated indices. Consider a (in particular, unitary or real) Drinfel'd twist F on H. Applying the results of Section 2.1 to H, X , Ξ, Ω and T we obtain the following: H F = U Ξ F is a Hopf ( * -)algebra, X is a left H F -module ( * -)algebra, while Ξ , Ω , T are H F -equivariant X -( * -)bimodules. The H F -actions are given by the -Lie derivative L ξ T := L F 1 ξ (F 2 T ) for all ξ ∈ H F and T ∈ T . On -vector fields X, Y ∈ Ξ , the -Lie derivative for all X ∈ Ξ and f, f ∈ X . 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. (25). This can be decomposed as T = p,r∈N 0 T p,r , where T 0,0 := X and for p + r > 0 In particular, for all T, T ∈ T , h, k ∈ X and ξ ∈ H F The third formula shows that ⊗ is actually ⊗ X , the tensor product over X . Let T ∈ T p,r . On any local chart (U, x) of M there unique functions T λ 1 ,...,λr Higher order differential forms are defined by the twisted skew-symmetrization of ⊗ ( -wedge product, an associative unital product), and we define Ω • := (Λ • Ω , ∧ ) to be the twisted exterior algebra of Ω (see [54] for more information).
The dual pairing , between vector fields and 1-forms can be equivalently considered as X -bilinear maps Ξ⊗Ω → X or Ω⊗Ξ → X ; for all arguments X ∈ Ξ, ω ∈ Ω these maps have the same images, which we respectively denote by the lhs and right-hand side (rhs) of the identity X, ω = ω, X . They have distinct twist deformations ( -pairings) defined by with (T, T ) = (X, ω) and (T, T ) = (ω, X) respectively. They satisfy for all ξ ∈ H F , X ∈ Ξ , ω ∈ Ω , (T, T ) = (X, ω) or (T, T ) = (ω, X), and h, h 1 , h 2 , h 3 ∈ X . Moreover, X, dh = L X h. As one can extend the ordinary pairing to higher tensor powers setting for all τ ∈ T p,r (the image will belong again to T p,r ) provided (T i , T i ) ∈ Ξ⊗Ω or (T i , T i ) ∈ Ω⊗Ξ for all i, so can one extend , to the corresponding twisted tensor powers using the same formula (41). Due to the 'onion structure' of (43) (i.e. the order of the T i and of the T i are opposite of each other), properties (42) are preserved, namely the -paring is H F -equivariant, as well as left, right and middle X -linear (if we chose a different order in (43) the deformed definition would need copies of R acting on the T i , T i ).

Twisted covariant derivatives and metrics
fulfilling, for all X, Y ∈ Ξ , h ∈ X , T, T ∈ T and ω ∈ Ω , Its curvature R F and torsion T F maps respectively act on all X, Y, Z ∈ Ξ through and are left X -linear maps They are in one-to-one correspondence with elements Setting F = 1⊗1 it follows that R = 1⊗1 and the definitions of twisted connection, torsion, curvature give the algebraic notion of connection, torsion, curvature of differential geometry. Consider a (classical) connection ∇ : Ξ⊗T → T on M and its equivariance Lie algebra e ⊆ Ξ (cf. [32]). The latter is a Lie subalgebra of the Lie algebra of vector fields defined by for all X, Y ∈ Ξ , T, T ∈ T and ω ∈ Ω (cf. [32] Proposition 2). A metric on M is a non-degenerate element g = g α ⊗g α ∈ (Ω⊗Ω)[[ν]] such that g = g α ⊗g α . We can view g as an element g = g A ⊗ g A ∈ Ω ⊗ Ω with g A ⊗g A = F 1 g α ⊗F 2 g α . A twisted connection ∇ F such that T F = 0 and ∇ F g = 0 is said to be a Levi-Civita (LC) connection for g. The associated Ricci tensor map and Ricci scalar of ∇ F are respectively defined by For a (pseudo-)Riemannian manifold (M, g) we define the Lie subalgebra and e the corresponding equivariance Lie algebra, we obtain k ⊆ e by the Koszul formula.
The following results are taken from [2,32].
] is a twist "based on Killing vector fields", then (52) defines a twisted LC connection ∇ F : Ξ ⊗ K[[ν]] T → T , and moreover for all X, Y ∈ Ξ . ∇ F is the unique LC connection with respect to g ; equivalently for all X, Y, Z ∈ Ξ . This twisted metric map g : Ξ ⊗ Ξ → X as well as the twisted curvature and Ricci tensor maps, are left X -linear in the first argument and right X -linear in the last argument. Also the twisted Ricci tensor map is in one-to-one correspondence with an element Ric F ∈ Ω ⊗ Ω such that Ric F (X, Y ) = X ⊗ Y, Ric F , by the non-degeneracy of the -pairing. The twisted curvature, Ricci tensor and Ricci scalar are U k F -invariant and coincide with their undeformed counterparts as elements

Twisted smooth submanifolds of R n of codimension 1
Here we collect the main results of [32] regarding a smooth submanifold M ⊂ D f ⊆ R n whose points x solve the single equation f (x) = 0. More generally, the solutions define a smooth manifold M c ; varying c we obtain a whole 1-parameter family of embedded submanifolds M c ⊆ R n of dimension n − 1. In [32] X stands for the * -algebra of smooth functions on D f , and also .. are understood in the smooth context.
Twist deformation of tangent and normal vector fields. According to Section 2.1 Ξ is a X -bimodule with X -subbimodules Ξ C , Ξ CC , Ξ t . We further define the X -bimodule By Proposition 9 in [32], the X -bimodules Ξ C , Ξ t and Ξ M =: Ξ C /Ξ CC are -Lie subalgebras of Ξ while Ξ CC is a -Lie ideal. Furthermore, we obtain the decomposition Ω ⊥ = X df = df X , and the twisted exterior algebras Ξ Let g = g α ⊗g α ∈ Ω⊗Ω be a (non-degenerate) metric on D f with inverse g −1 = g −1α ⊗g −1 α .
are the X -bimodules of normal vector fields and tangent differential forms. The open subset where the restriction g −1 . From now on we denote the restrictions of Ξ, Ξ t , Ξ ⊥ , Ω, Ω t , Ω ⊥ to D f by the same symbols and by k ⊆ Ξ t the Lie subalgebra of Killing vector fields with respect to g which are also tangent to M c ⊆ D f . The deformed analogues of (61) ] then by Proposition 10 in [32], there are direct sum decompositions into orthogonal X -bimodules, with respect to g and g −1 respectively. Ξ t is a -Lie subalgebra of Ξ , Ω t and Ξ ⊥ are orthogonal with respect to the -pairing and actually Ω t = {ω ∈ Ω | Ξ ⊥ , ω = 0}. Furthermore, the restrictions ]-modules; and similarly for their -tensor (and -wedge) powers. The orthogonal projections pr t : Ξ → Ξ t , pr ⊥ : Ξ → Ξ ⊥ , pr t : Ω → Ω t and pr ⊥ : Ω → Ω ⊥ and their (unique) extensions to multivector fields and higher rank forms are the C[[ν]]-linear extensions of their classical counterparts. They, as well as and Ω CC := Ω f = f Ω , we further obtain The following proposition assures that every element of Ξ M can be represented by an element in Ξ t and every element of Ω M can be represented by an element in Ω t .
Let ∇ be the LC connection corresponding to (D f , g) and ∇ F be the twisted LC connection corresponding to g . The induced twisted second fundamental form and LC connection on the family of submanifolds M c are (48). We now summarize results of Propositions 3, 12 and 13 in [32]. As g F t = g t , also the twisted second fundamental form, curvature, Ricci tensor and Ricci scalar on M are U k F -invariant and coincide with the undeformed ones as elements , are U k F -equivariant maps, and for all X, Y, Z ∈ Ξ t they actually reduce to where F 1 ⊗F 2 ⊗F 3 is the inverse of (18); these maps are left (resp. right) X -linear in the first (resp. last) argument, 'middle' X -linear otherwise, in the sense g t (X h, Y ) = g t (X, h Y ), etc. Furthermore, the following twisted Gauss equation holds for all X, Y, Z, The twisted first and second fundamental forms, Levi-Civita connection, curvature tensor, Ricci tensor, Ricci scalar on M are finally obtained from the above 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, one can do the same on any other M c .
Decompositions (63) in terms of bases or complete sets. In terms of Cartesian coordinates (x 1 , . . . , x n ) of R n the components of the metric and of the inverse metric on R n are denoted by g ij = g(∂ i , ∂ j ) and g ij = g −1 (dx i , dx j ) (as before ∂ i ≡ ∂/∂x i ). Using them we lower and raise indices: Let (see Proposition 8 in [32]); these relations hold also without . The projection pr ⊥ (C[[ν]]linear extension of pr ⊥ ) on X ∈ Ξ , ω ∈ Ω can be equivalently expressed as (see Proposition 14 in [32]). By the -bilinearity of g these equations imply in particular in terms of the left and right decompositions ω One can decompose df, N ⊥ , θ, U ⊥ themselves in the same way, if one wishes. If the metric is Euclidean (g ij = δ ij ) or Minkowski [g ij = g ij = η ij = diag(1, ..., 1, −1)] one makes (73) more explicit replacing Finally, we can express the tangent projection acting on X ∈ Ξ , ω ∈ Ω simply as pr t (X) = Having determined bases of Ξ ⊥ , Ω ⊥ we now consider Ξ t , Ω t . The globally defined sets Θ t := ϑ j n j=1 , S W := {W j } n j=1 , where ϑ j := pr t (dx j ), W j := pr t (∂ j ) =: K V j , are respectively complete in Ω t , Ξ t ; they are not bases, because of the linear dependence relations ϑ j f j = 0, f j W j = 0. An alternative complete set (of globally defined vector fields) in Ξ t is In fact, L ij manifestly annihilate f , and S L is complete because the combinations Kf i L ij = W j make up S W . Clearly L ij = −L ji , so at most n(n−1)/2 L ij (e.g. those with i < j) are linearly independent over R (or C), while S L is of rank n−1 over X because of the dependence relations (square brackets enclosing indices mean a complete antisymmetrization of the latter). Contrary to the W j , the L ij are anti-Hermitian under the * -structure L * ij = −L ij and do not involve g, so they can be used even if we introduce no metric. Setting By the mentioned propositions, every complete set of Ω t , e.g. Θ t , is also a complete set of Ω t ; similarly, every complete set of Ξ t , e.g. S W or S L , is also a complete set of Ξ t .
3 Twisted algebraic submanifolds of R n : the quadrics We can apply the whole machinery developed in the previous chapter to twist deform algebraic manifolds of codimension 1 embedded in R n provided we adopt X = Pol • (R n ), etc. everywhere. We can assume without loss of generality that the f be an irreducible polynomial function 5 . 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 (or C), 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. If M c are manifestly symmetric under a Lie group 6 K, then such a g exists and contains the Lie algebra k of K (if M is maximally symmetric then k is even complete -over X -in Ξ t ). In general, given any set S of vector fields that is complete in Ξ t the question is whether there are combinations of them (with coefficients in X ) that close a finite-dimensional Lie algebra g.
Here we answer this question in the simple situation where the L ij themselves close a finite-dimensional Lie algebra g. This means that in (77) 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 . In the next subsection we find some results valid for all n ≥ 3 drawing some general consequences from the only assumptions X = Pol • (R n ) and g ⊂ aff(n); in particular, in sections 3.1, 3.2 we show that the global description of differential geometry on R n , M c in terms of generators and relations extends to 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). 6 For instance, the sphere their twist deformations, in such a way to preserve the spaces consisting of polynomials of any fixed degrees in the coordinates x i , differential dx i and vector fields chosen as generators. In section 4 we shall analyze in detail the twisted quadrics embedded in R 3 .
If f is of degree two then there are real constants a µν = a νµ (µ, ν = 0, 1, ..., n) such that (77) has already the desired form i.e. the L ij span a finite-dimensional Lie algebra g over R. This is a Lie subalgebra of the affine Lie algebra of R n , because all L ih act as linear transformations of the coordinates x k : Let r := rank(a µν ). To identify g for irreducible f 's (r > 2) 7 we note that by a suitable Euclidean transformation (this will be 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, and coordinates are ordered so that with l ≤ m ≤ n; moreover, if m < n one can make a 00 = 0 by translation of a x j with j > m.
The associated new L ij (which are related to the old by a linear transformation) fulfill It is easy to check that r = n+1 if m = n, r = m+2 if m < n. One can always make a 1 = 1 by replacing f → f /a 1 ; one can make also the other nonzero a i 's in (82) be ±1 by the rescalings x i → y i := |a i | 1/2 x i of the corresponding coordinates (another affine transformation). So the associated new L ij fulfill (82) with the a i ∈ {−1, 0, 1}. Then: Lie subalgebra, the radical R(g) (the largest solvable ideal) of g.
• Finally, the L ij with i < j ≤ m make up a basis of a m(m − 1)/2-dimensional simple Lie-subalgebra g s so(l, m−l), in view of the signs of a i , a j .
Summing up, the Levi decomposition of g becomes g so(l, m−l) × R.
The cones, which in the y coordinates are represented by the homogeneous equations strictly speaking are not encompassed in the above analysis because the Jacobian matrix (f i )(y) vanishes at the apex y = 0 (the only singular point). They are algebraic varieties that are limits of the hyperboloids f c (y) = 0 as c → 0. If we omit the apex, a cone becomes a disconnected union of two nappes (which are open in R n ), and g is spanned not only by the L ij , but also by the central anti-Hermitian element η = x i ∂ i + n/2 generating dilatations; note that all of them vanish on the apex. Hence g so(l, n−l) × R in this case.
If we endow R n with the Euclidean metric, the metric matrix g ij = δ ij is not changed by the above Euclidean changes of coordinates, because the Euclidean group is the isometry group H of R n , whereas its nonzero (diagonal) elements are rescaled if we rescale x i → |a i | 1/2 x i . Similarly, if we endow R n with the Minkowski metric, Euclidean changes of coordinates involving only the space ones, or a translation of the time coordinate, do not alter the metric matrix g ij = η ij .

Twisted differential calculus on R n by generators, relations
Let us abbreviate ξ i := dx i . We name differential calculus algebra on R n the unital associative * -algebra Q • over C generated by Hermitian elements {1, The x 0 ≡ 1, x i , ξ i , ∂ i play respectively the role of the unit, of Cartesian coordinate functions on R n , of differentials dx i of x i , of partial derivatives ∂/∂x i with respect to x i . This is the adaptation of the definition of Q • in the smooth context (sections 3.1.3, 3.2.3 in [32]) to the polynomial one: the relations in the first two lines define the algebra structure of X , the other ones determine the relations (113-114) of [32] for the current choice of X and of the pair {ξ i }, {∂ i } of dual frames. The x µ (µ = 0, ..., n) span the fundamental module (M, τ ) of U aff(n) (the invariant element 1 itself spans a 1-dim, non-faithful submodule), the ξ i span a related module (M, τ ), the ∂ i the contragredient one (M ∨ , τ ∨ ). More precisely they are related by the first relation and g x 0 = x µτ µ0 (g) implyτ µ0 (g) = ε(g)δ µ0 . We encompass these U aff(n)modules into a single one ( M, ρ) spanned by (a 0 , a 1 ,..., a 3n ) ≡ (1, x 1 , ..., x n , ξ 1 , ..., ξ n , ∂ 1 , ..., ∂ n ). All are trivially also U g-modules; also g is, under the adjoint action. Of course, this U aff(n) action is compatible with the relations (83-84); the ideal I generated by their left-hand sides in the free * -algebra A f generated by {a 0 , a 1 , ..., a 3n } is U aff(n)-invariant. The U aff(n)-action is also compatible with the invariance of the exterior derivative, because g ξ i = d(g x i ).
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 so far 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).
When choosing a basis B of Q • made out of monomials in these generators, relations (83-84) allow to order them in any prescribed way; in particular we may choose (we define β 0, 0, 0 := 1). The * -algebra structure of Q • is compatible with the form grading and the one defined by β p, q, r = q − r (p, q, r are the total degrees in ξ i , x i , ∂ i respectively).
Fixing part or all of p, q, r we obtain the various relevant U aff(n) modules or module subalgebras or X -bimodules: Λ • , Λ p , Ω • , Ω p , .... For instance the exterior algebra Λ • is generated by the ξ i alone (q = r = 0) and its = p component is the U aff(n)-submodule of exterior p-forms Λ p ; by (84) 3 dim(Λ p ) = n p ; in particular this is zero for p > n, 1 for p = n, and Λ • = n p=0 Λ p . Let X q be the component of X of degree q, and X q := q h=0 X q (i.e. X q , X q consist resp. of homogeneous and inhomogenous polynomials in x i of degree q); X = ∞ q=0 X q is trivially a filtered algebra X = ∞ q=0 X q . Let D be the unital subalgebra generated by the ∂ i alone, D r its component of degree r, and D r := r h=0 D r ; then D = ∞ r=0 D r is trivially a filtered algebra D = ∞ r=0 D r . Finally, let By (85) the U aff(n) action maps Λ p , X q , D r into themselves, and all Q pqr are U aff(n)- *modules. By (83-84), D r X q = X q D r , whence (this multiplication rule would not hold if we had defined Q pqr := Λ p X q D r , because, D r X q = r i ≤ r}. Q • is graded by p and filtered by both q, r; it decomposes as Choosing a twist F based on U aff(n) (in particular, on U g) and setting (10) for all a, b ∈ Q • one makes Q • into a U aff(n) F -module (resp. U g F -module) algebra Q • with grading (whereas the grading is not preserved). In the appendix we prove Proposition 2 The vector fields ∂ i := S(β) ∂ i = τ ij (β)∂ j are the -dual ones to the ξ i = dx i ; under the U aff(n) (and U g) action they transform according to g ∂ i = τ ij [S F (g)]. The polynomials relations (83-84) are deformed into the ones where R µν ij := (τ µi ⊗τ νj )(R). Defining Q pqr := Λ p X q D r , we find not only hold as equalities of C[[ν]]-modules. A basis B pqr of Q pqr is obtained replacing all products in the definition of B pqr by -products. Q • is graded by p, filtered by both q, r, and Q • is a U g F -module * -algebra with the Q pqr as * -submodules, if F is either real or unitary; correspondingly the involution is the undeformed one * , respectively is given by (29), i.e.
In the Q • framework X h = (R 1 h) (R 2 X) + X (h), while so far it stood just for theproduct of the vector field X by the function h from the right, i.e. for the first term at the rhs; denoting the latter by X h : The U aff F -equivariant relations (90-91) defining Q • have the same form (see e.g. formulae (1.10-15) in [27]) as the quantum group equivariant ones defining the differential calculus algebras on the celebrated 'quantum spaces' introduced in [23]. The relations, among (90-91), that involve only the generators x i , ∂ j of the twisted Heisenberg algebra on R n (the p = 0 component of Q • ) were already determined in [24,25], while (92) extends results of [29].

Twisted differential calculus on M by generators, relations
and the remaining relations of the type eq. (113) in [32], i.e.
with suitable t a l , C γ αβ ∈ X . For instance, if S ≡ {L ij , V ⊥ } then the dependence relations in the first line amount to (76), while the commutation relations in the second line have constant C γ αβ and amount to (79) for α, β ≤ B. We collectively rename 1, x 1 , ..., x n , ξ 1 , ..., ξ n , e 1 , ..., e B as a 0 , a 1 , ..., a N ; we denote as A • the free algebra generated by a 0 , ..., a N , and as A pqr the subspace consisting of polynomials in the a A of degree q in the x i , of degree r in the e α and homogeneous of degree p in the ξ i . Clearly A • is graded by p and filtered by both q, r; it decomposes as For all c ∈ R denote as {f J c (a 0 , ..., a N )} J∈J the set of polynomial functions at the lhs of (96), (83) , (95) involving only e α with α ≤ B, together with which are (78) and its exterior derivative. Let I Mc be the ideal generated by all the f J c (a) in A • . We define the differential calculus algebra on M c as the quotient By (85), (96) 2 the a i span a (reducible) U g- * -module. Hence A • , which is generated by them, is a U g-module * -algebra, and the A pqr are U g- * -submodules. It is immediate to check that also the f J c (a) span a (reducible) U g- * -module, more precisely g f c = ε(g)f c , while more generally g f J c (a) is a numerical combination of the f J c (a) appearing in the same equation where f J c (a) appears, e.g. g (ξ i ξ j + ξ j ξ i ) = (ξ h ξ k + ξ k ξ h )τ hi (g (1) )τ kj (g (2) ). Therefore I Mc is a U g- * -module, and Q • Mc is a U g-module * -algebra as well; moreover I pqr Mc  := A • /I Mc defines a U g F -module * -algebra, which we shall name twisted differential calculus algebra on M c ; taking the quotient commutes with deforming the product: All components Q pqr is graded by p, filtered by both q, r, and as * -submodules. If F is real the involution is undeformed * . If F is unitary the involution is given by (29), i.e. on ξ i , x i * acts as in (94), while L * ij = −τ ih β (1) τ jk β (2) L hk (this differs from L * ij = −L ij ).
These results are the strict analogue of their undeformed counterparts. Relation (106) In section 4 we explictly determine all of the relationsf J c (a ) = 0 in the specific case of some deformed quadrics in R 3 .

The quadrics in R 3
Using the notions and results presented in the previous sections, here we study in detail twist deformations of the quadric surfaces in R 3 . As usual, we identify two quadric surfaces if they can be translated into each other via an Euclidean transformation. This leads to nine classes of quadrics, identified by their equations in canonical (i.e. simplest) form. These are summarized in Fig. 1, together with their rank, the associated symmetry Lie algebra g, and the type of twist deformation we perform. A plot of each class is given in Fig. 2. These classes make up 7 families of submanifolds, differing by the value of c. In fact classes (f), (g), (h) altogether give a single family: (f) consists of connected manifolds, the 1-sheeted hyperboloids; (g), (h) of two-component manifolds, the 2-sheeted hyperboloids and the cone, which has two nappes separated by the apex (a singular point); all are closed, except the cone. For all families, except (i) (consisting of ellipsoids), we succeed in building U g-based Drinfel'd twists of either abelian (30) or Jordanian (31) type (depending on the coefficients of the normal form) and through the latter in creating explicit twist deformations. Those twists are the simplest ones resp. based on an abelian or "ax + b" Lie subalgebra of the symmetry Lie algebras. Note that there are other choices of Drinfel'd twists on the "ax + b"-Lie algebra. In particular we like to mention the twist of Theorem 2.10 of [34], which is the real (i.e. F * ⊗ * = (S⊗S)[F 21 ]) counterpart of the unitary Jordanian twist we utilize; both twists lead to the same commutation relations. Since we are especially interested in describing the deformed spaces in terms of deformed generators and relations, i.e. we intend to explicitly calculate -commutators and the twisted Hopf algebra structures, we use abelian and Jordanian twists, which admit an explicit exponential formulation. Furthermore, all of the considered symmetry Lie algebras (except the one of the ellipsoids) contain an abelian or "ax + b" Lie subalgebra, which allows us to perform a homogeneous deformation approach for all quadric surfaces. We devote a subsection to each of the remaining six families of quadrics, and a proposition to each twist deformation; propositions are proved in the appendix. Throughout this section the star product X h of a vector field X by a function h from the right is understood in the Q , Q Mc sense (see section 3.1) For every fixed b, {M c } c∈R is a foliation of R 3 . The Lie algebra g is spanned by the vector fields L 12 = x 1 ∂ 2 , L 13 = x 1 ∂ 3 + b∂ 1 , L 23 = b∂ 2 , which fulfill Clearly, g h(1), the Heisenberg algebra. The actions of the L ij on the x h , ξ h , ∂ h are Proposition 4 F = exp(iνL 13 ⊗ L 23 ) is a unitary abelian twist inducing the following twisted deformations of U g, of Q • on R 3 and of Q • Mc on the parabolic cylinders (107). The U g F counit, coproduct, antipode on the {L ij } 1≤i<j≤3 coincide with the undeformed ones, except The twisted star products and Lie brackets of the L ij coincide with the untwisted ones. The twisted star products of the L ij with the x i , ξ i ≡ dx i , ∂ i , and those among the latter, equal their undeformed counterparts, except L 12 x 2 = L 12 x 2 − iνb L 23 , Hence the -commutation relations of the U g F -equivariant * -algebra Q • read In terms of star products L 12 = x 1 ∂ 2 , L 13 = x 1 ∂ 3 + b∂ 1 , L 23 = b∂ 2 . Also the relations characterizing the U g F -equivariant * -algebra Q • Mc , i.e. equation (107), its differential and the linear dependence relations, keep the same form: Alternatively, one could twist everything by the unitary abelian twist F = exp(iνL 12 ⊗ L 23 ).

(b)
Family of elliptic paraboloids: a 2 > 0, a 3 = 0, a 03 < 0 Their equations in canonical form are parametrized by a = a 2 , c = −a 00 ∈ R, b = −a 3 ∈ R + and read (e) One-sheet elliptic hyperboloid with a 1 = 1 2 , a 2 = −a 3 = 2 Clearly, g so(2)×R 2 . The actions of the L ij on the x h , ξ h , ∂ h are given by Proposition 5 F = exp(iνL 13 ⊗ L 23 ) is a unitary abelian twist inducing the following twisted deformation of U g, of Q • on R 3 and of Q • Mc on the elliptic paraboloids (113). The U g F counit, coproduct, antipode on the {L ij } 1≤i<j≤3 coincide with the undeformed ones, except (117) The twisted star products and Lie brackets of the {L ij } 1≤i<j≤3 coincide with the untwisted ones except L 12 L 12 = L 2 12 +iνaL 23 L 13 . The twisted star products of the L ij with the x i , ξ i ≡ dx i , ∂ i , and those among the latter, equal their undeformed counterparts, except L 12 u 3 = L 12 u 3 − iνa L 23 u 2 , u 3 L 12 = u 3 L 12 + iνa u 1 L 13 , where u i = x i , ξ i . Hence the -commutation relations of the U g F -equivariant algebra Q read while those among the tangent vectors L ij and the generators x i , ξ i , ∂ i read (120) In terms of star products Also the relations characterizing the U g F -equivariant * -algebra Q • Mc , i.e. equation (113), its differential and the linear dependence relations keep the same form

(c)
Family of elliptic cylinders: a 2 > 0, a 3 = a 0i = 0, a 00 < 0 Their equations in canonical form are parametrized by c, a ≡ a 2 ∈ R + and read This is essentially the same as Proposition 15 in [32]. 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; the relevant relations are (115) b=0 , We correspondingly adopt the unitary abelian twist F = exp(iν∂ 3 ⊗ L 12 ).
Proposition 16 in [32]. F = exp(iν∂ 3 ⊗L 12 ) is a unitary abelian twist inducing the following twist deformation of U g, of Q • on R 3 and of Q • Mc on the elliptic cylinders (122). The U g F counit, coproduct, antipode on {∂ 3 , L 12 } coincide with the undeformed ones. The twisted star products and Lie brackets of {∂ 3 , L 12 } coincide with the untwisted ones. The twisted star products of ∂ 3 , L 12 with x i , ξ i ≡ dx i , ∂ i , and those among the latter, equal the untwisted ones, except Hence the -commutation relations of the U g F -equivariant algebra Q read In terms of star products L 12 = x 1 ∂ 2 − ax 2 ∂ 1 . Also the relations characterizing the U g F -equivariant * -algebra Q • Mc , i.e. eq. (122), its differential and (76), keep the same form:

The deformation via the abelian twist
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 7. Eq. (128-129) 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, by Theorem 7 in [2]. Furthermore, for all X, Y ∈ S t , leading to the same principal curvatures κ 1 = 0, κ 2 = 1/R, Gauss and mean curvatures as in the undeformed case.
For all fixed a, b > 0, {M c } c∈R is a foliation of R 3 . The Lie algebra g is spanned by the vector fields To compute the action of F on functions it is convenient to adopt the eigenvectors of H as new coordinates. In fact, H y i = λ i y i with λ 1 = 2, λ 2 = −2 and λ 3 = 0. Abbreviating ∂ i = ∂ ∂y i , the inverse coordinate and the partial derivatives transformations read In the new coordinates f c (y) = 1 2 y 1 y 2 − by 3 − c and H = 2(y 1∂ 1 − y 2∂ 2 ), E = y 1∂ 3 + 2b∂ 2 , E = y 2∂ 3 + 2b∂ 1 . The actions of H, E, E on coordinate functions, differential forms η i = dy i and vector fields are given by for all 1 ≤ i ≤ 3 The * -structures on U g F , Q • , Q • Mc remain undeformed apart from (y 2 ) * = y 2 + 2iνb and (∂ 1 ) * = −∂ 1 + iν∂ 3 . The twisted star products of {H, E, E } coincide with the untwisted ones, except The twisted star products of H, E, E with y i , η i ,∂ i equal the untwisted ones, except (139) the twisted star products among y i , η i ,∂ i equal the untwisted ones, except Hence the -commutation relations of the U g F -equivariant algebra Q read y 1 y 2 = y 2 y 1 − 2biνy 1 , and In terms of star products 4.5 (e) Family of hyperbolic cylinders: a 2 < 0, a 3 = a 0µ = 0 Their equations in canonical form are parametrized by c, a ≡ −a 2 ∈ R + and read For every a > 0, this equation with c = 0 singles out a variety π consisting of two planes intersecting along the z-axis; {M c } c∈R + is a foliation of R 3 \ π. The case c < 0 is reduced to the case c > 0 by a π/2 rotation around the z-axis. Eq. (144) can be obtained from the one (131) characterizing the hyperbolic paraboloids (d) setting b = 0. Hence also the tangent vector fields L ij (or equivalently H, E, E ), their commutation relations, their actions on the x h , ξ h , ∂ h (or equivalently on the y h , η h = dy h ,∂ h defined by (134-135)), the commutation relations of the L ij with the x h , ξ h , ∂ h can be obtained from the ones of case (d) by setting b = 0. The L ij fulfill again (132), or equivalently (133), so that g so(1, 1)×R 2 .
Proposition 8 F = exp(iνL 13 ⊗ L 23 ) is a unitary abelian twist inducing the twisted deformation of U g, of Q • on R 3 and of Q • Mc on the hyperbolic cylinders (144) that is obtained by replacing a → −a in Proposition 16 in [32], section 4.3.
We can also deform everything with the same Jordanian twist as in (d). We find 4.6 (f-g-h) Family of hyperboloids and cone: a 2 , −a 3 > 0 Their equations in canonical form are parametrized by a = a 2 , 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 For all a, b > 0, {M c } c∈R\{0} is a foliation of R 3 \ M 0 , where M 0 is the cone of equation f 0 = 0 (see section 4.6.2). 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 the eigenvectors of H as new coordinates; the eigenvalues are λ 1 = 2, λ 2 = 0 and λ 3 = −2. Abbreviating η i := dy i ,∂ i := ∂/∂y i ,∂ 2 :=∂ 2 ,∂ 1 := 2a∂ 3 ,∂ 3 := 2a∂ 1 the inverse coordinate and the partial derivative transformations read In the new coordinates, ∂ i * = −∂ i , f c (y) = 1 2 y 1 y 3 + a 2 (y 2 ) 2 − c and The actions of H, E, E on any u i ∈ {y i ,∂ i , η i } read Proposition 17 in [32]. F = exp(H/2 ⊗ log(1 + iνE)) is a unitary twist inducing the following twisted deformation of U g, of Q • on R 3 and of Q • Mc on the hyperboloids or cone (145). The U g F coproduct, antipode on {H, E, E } are given by The twisted star products of {H, E, E } coincide with the untwisted ones, except The twisted star products of u i = y i , η i ,∂ i with v j = y j , η j ,∂ j and with H, E, E are given by Hence the -commutation relations of the U g F -equivariant algebra Q read as follows: for u i = y i ,∂ i ; the twisted Leibniz rule for the derivatives read √ a y 1 ∂ 3 + 2ν 2 y 1 ∂ 2 , ∂ 2 y 2 = 1 + y 2 ∂ 2 ,∂ 1 y 3 = 2a + y 3 ∂ 1 + i2ν √ a y 2 ∂ 1 + 2ν 2 y 1 ∂ 1 , while the twisted wedge products fulfill The -commutation relations between generators of Q and the tangent vectors H, E, E are In terms of star products The relations characterizing the U g F -equivariant * -algebra Q •

Circular hyperboloids and cone embedded in Minkowski R 3
We now focus on the case This covers the circular cone and hyperboloids of one and two sheets. We endow R 3 with the Minkowski g is equivariant with respect to U g, where g so(2, 1) is the Lie * -algebra spanned by the vector fields L ij , tangent to M c = f −1 c ({0}). The first fundamental form g t := g • (pr t ⊗pr t ) makes M c Riemannian if c < 0, Lorentzian if c > 0, whereas is degenerate on the cone M 0 . Moreover, where V ⊥ = f j η ji ∂ i = x i ∂ i (outward normal); in particular, this implies the proportionality relation II(v α , v β ) = − 1 2c g αβ V ⊥ (here g αβ := g(v α , v β )) between the matrix elements of II, g t in any basis S t := {v 1 , v 2 } of Ξ t , and, applying the Gauss theorem, one finds the following components of the curvature and Ricci tensors, Ricci scalar (or Gauss curvature) on M c : [we recall that by the Bianchi identity one can express the whole curvature tensor on a (pseudo)Riemanian surface in terms of the Ricci scalar in this way, and that R t . All diverge as c → 0 (i.e. in the cone M 0 limit). M c is therefore de Sitter space dS 2 if c > 0, the union of two copies of anti-de Sitter space AdS 2 (the hyperbolic plane) if c < 0. In appendix B.7.2 we recall how these results can be derived. In terms of the y i coordinates and the tangent vector fields H, E, E (145), (76) become the linear dependence relations y 1 y 3 +(y 2 ) 2 = 2c and y 3 E −y 1 E −y 2 H = 0, i.e. (160) for a = 1, ν = 0. At all points of M c at least two out of E, E , H are non-zero (in the case c = 0 we have already excluded the only point where this does not occur, the apex) and make up another basis S t = { 1 , 2 } of Ξ t . More precisely, we can choose 1 := E, 2 := E in a chart where y 2 = 0, 1 := E, 2 := H in a chart where y 1 = 0, 1 := E , 2 := H in a chart where y 3 = 0. One can use (161), (162) with each basis S t ; g αβ stands for g αβ ≡ g( α , β ), and these matrix elements are given in (182). Alternatively, we can use the complete set S c t = {E, E , H} on all of M c , keeping in mind the mentioned linear dependence relations.
We now analyze the effects on the geometry of the twist deformation of Proposition 17 in [32] restated above. The curvature (and Ricci) tensor on R 3 remain zero. Moreover, eq. (66), (67) apply; namely, on M c the first and second fundamental forms, as well as the curvature and Ricci tensor, remain 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 (compare also to [2] Theorem 7 and eq. 6.138). Also the Ricci scalar (or Gauss curvature) R F the second equality holds because V ⊥ is U k-invariant. Similarly, by (67), (55) for all X, Y, Z ∈ Ξ t ; the twisted counterpart of (162) is obtained choosing (X, Y, Z) = (v α , v β , v γ ). Hence the matrix elements of II F , R F t , Ric F t in any basis S t are obtained from those of the twisted metric g t on M c . In the appendix we sketchily prove that on E, E , H Finally, we also show that the twisted Levi-Civita connection on E, E , H gives We recall that a sheet of the hyperboloid M c , c < 0, is equivalent to a hyperbolic plane. Other deformation quantizations of the latter have been done, in particular that of [8] in the framework [52,7] (cf. the introduction). However, while the -product [8] is U k-equivariant, i.e. relation (16) (which is the 'infinitesimal' version of the invariance property (10) in [8] or (1) of [7]) holds, our -product is U k F -equivariant i.e. relation (21) holds.

Additional twist deformation of the cone (h)
The equation of the cone M 0 in canonical form is (145) with c = 0. In addition to the tangent vector fields L ij or H, E, E fulfilling (146) also the generator D = x i ∂ i = y i∂ i of dilatations is tangent to M 0 (only), D ∈ Ξ M 0 , since D(f ) = 2f ; furthermore it commutes with all L ij . Hence the anti-Hermitian elements H, E, E , D span a Lie algebra g so(2, 1) × R. The actions of H, E, E on Q M are as in cases (e-f), while that of D is determined by Therefore, we can build also abelian twist deformations of M 0 of the form F = exp(iνD ⊗ g), g ∈ g.
Here we choose g = L 13 . The cases with L 23 , L 12 are similar. Setting µ 1 = 1 = −µ 3 and µ 2 = 0, for u i , v i ∈ {y i , η i } we find Having this in mind, in the appendix we easily determine the twist deformed structures.
Proposition 10 F = exp(iνD ⊗H/2) is a unitary abelian twist inducing the following twisted deformation of U g, of Q • on R 3 and of Q • Mc on the cone M 0 . The U g F counit, coproduct, antipode on {D, H, E, E } coincide with the undeformed ones, except The twisted star products among D, L ij coincide with the untwisted ones. The twisted star products of D, L ij with u i ∈ {y i , η i },∂ i coincide with the untwisted ones, except The twisted star products among y i , η i ,∂ i read Hence the -commutation relations of the U g F -equivariant algebra Q are The * -structures on U g F , Q • , Q M are undeformed, except which are nontrivial for i = 1 and i = 3. In terms of star products and the relations characterizing the U g F -equivariant * -algebra Q • Mc , i.e. equation (145) c=0 , its differential and the linear dependence relations become One may wonder whether the irreducibility in R[x 1 , . . . , x n ] of each polynomial f 1 , . . . , f k is a sufficient condition in order that Q lies in (f 1 , . . . , f k ), the ideal generated by f 1 , . . . , f k . The following example answers in the negative.
Example 11 Consider in R 3 the variety defined by the system where the first equation represents a cubic cylinder C. Since the curve defined by is smooth in P 2 C , the cylinder C is smooth and the polynomial 2x 3 − y 3 − 1 is irreducible in R[x, y, z] (the same conclusion is obvious for y − 1). The real variety defined by (173) is the line which is obviously smooth. Furthermore, the equation of the tangent plane to the cylinder C at the point (1, 1, t) ∈ l is 2(x − 1) − (y − 1) = 0, hence the intersection is transversal at each point of l. On the other hand, the plane π defined by x + y − 2 = 0 contains l but x + y − 2 ∈ (2x 3 − y 3 − 1, y − 1), since both 2x 3 − y 3 − 1 and y − 1 do vanish at the points (exp 2 3 πi, 1, t), ∀t ∈ R, and conversely x + y − 2 does not.
In view of the previous example, it is interesting to ask for some sufficient condition in order that Q ∈ (f 1 , . . . , f k ). An answer is provided by Theorem 1, which we now prove.  . . . , f k ). Finally, for a complex-valued h = Q 1 + iQ 2 vanishing on M both Q 1 , Q 2 do, and therefore h belongs to the complexification of (f 1 , . . . , f k ).
As for the last statement, the projective closure X ⊆ P n C of the zero locus of (1) in P n C has degree at least s. On the other hand, s is the maximum degree so X is a complete intersection in P n C . Then, there cannot be other components and the variety defined by (1)  Using the definition β := F 1 · S(F 2 ), a a = (R 1 a) (R 2 a ) and the relation valid for all triangular Hopf algebras, one can prove relations (90-91) as follows: obtained from the previous one applying d By (85) the action of either leg F 1 , F 2 of the twist, or F 1 , F 2 of its inverse, as well as of any tensor factor in the (iterated) coproducts of F 1 , F 2 , F 1 , F 2 , maps every homogeneous polynomial in ξ i or ∂ i into another one of the same degree, and every polynomial in x i into another one of the same degree: hence (92), (93) follow. Finally, the relations g ∂ i = τ ij [S F (g)]∂ j , (94) are straightforward consequences of (19), (29), (85):

B.2 Proof of Proposition 3
All statements up to (106) and the statement that the -polynomialsf J c (a ) have the same degrees in x i , ξ i , e α as the polynomials f J c (a) are straightforward consequences of (85) and of what precedes the proposition. Under U g the f i transform as the ∂ i ; in fact, since g f = ε(g)f , we find this can be computed more explicitly using the relation (see e.g. eq. (126) in [29])

B.3 (a) Family of parabolic cylinders
Proof of Proposition 4 Since L 13 and L 23 are commuting anti-Hermitian vector fields it follows that F is a unitary abelian twist on U g. We find S(β) = exp(−iνL 12 L 23 ), and where in the last equation we use F(1 ⊗ L 12 ) = (1 ⊗ L 12 )F since the second leg of the twist is central. Moreover F(L 13 ⊗ 1) = ∞ n=0 (iν) n n! L n+1 13 ⊗ L n 23 = (L 13 ⊗ 1)F and F(L 23 ⊗ 1) = (L 23 ⊗ 1)F show that ∆ F (L 13 ) = ∆(L 13 ) and ∆ F (L 23 ) = ∆(L 23 ). We have thus proved the claimed coproducts ∆ F (L ij ). Next, the latter and the antipode property µ[(S F ⊗id)∆ F (g)] = (L ij )1 = 0 easily determine the twisted antipode S F (L 12 ) as in (110), and other ones S F (L ij ) = S(L ij ) = −L ij . Furthermore, since the L 23 contained in the second leg of the twist commutes with L k we conclude that L ij L k = L ij L k and [L ij , L k ] = [L ij , L k ] for all 1 ≤ i < j ≤ 3 and 1 ≤ k < ≤ 3. For the same reason one gets for all 1 ≤ i, j, k ≤ 3 On the other hand by (109) we obtain for all 1 ≤ i, j, k ≤ 3. The commutation relations respectively follow. Furthermore this means we can express the generators of the Lie algebra in terms of the twisted module action, namely hold. Again by (109) L 12 L 23 x i = 0, L 12 L 23 ξ i = 0, L 12 L 23 ∂ i = 0 for all i = 1, 2, 3, which implies that the * -structure on Q • remains the same as on Q • Mc : * = * .

B.4 (b) Family of elliptic paraboloids
Proof of Proposition 5 Since L 13 , L 23 are commuting anti-Hermitian vector fields F is a unitary abelian twist on U g. for n > 0, which follow by iteratively applying eq.(114). Then imply (117) 1 . The twisted antipodes (117) 2 follow using the properties µ The twisted tensor and star products coincide with the untwisted ones as soon as one of the factors is L 13 or L 23 . This is because the latter commute with both legs of the twist. Among all star products of generators of g only the one The anti-Hermitian vector fields H and E satisfy eq.(133), which implies that F = exp(H/2 ⊗ log(1 + iνE)) is a unitary Jordanian twist on U g. We note that for all m > 0, which follows by iteratively applying (133). In particular this implies where we have made use of the expansions Both are well-defined in the ν-adic topology. Applying this result iteratively we obtain and F(1⊗E ) = (1⊗E )F, which determine ∆ F (E ) as in (137). Next, it is straightforward to check that the coproducts ∆ F (g), with g = H, E, E , and the antipode property µ[(S F ⊗ id)∆ F (g)] = (g)1 = 0 determine the twisted antipodes S F (g) as in (137).
To compute the twisted tensor and star products we first make only the first leg F 1 of F to act on its eigenvectors H, E, E , u i , ∂ i (generators of U g and of Q), and find for all u i ∈ {y i , η i }, 1 ≤ i ≤ 3; note that the exponents ±λ i /2 take the values ±1, 0. This simplifies the computation of the action of the second leg F 2 on the second factor; using (133) and (136) and noting that only the terms of degree lower than two in the power expansion of 1/(1 + iνE) contribute to its action on the H, E, E , u i , ∂ i , by a direct computation one thus finds the star products (138-140). In particular the twisted tensor and star products are trivial if H, u 3 or∂ 3 appears in the first factor. The twisted commutation relations (141)-(142) and the twisted submanifold constraints (143) follow. For the twisted star involution note that since E y i = δ i 3 y 1 + 2bδ i 2 and λ 3 = 0, λ 2 = −1, while E ∂ i = −δ i1∂3 , and λ 1 = 1.
To compute the twisted tensor and star products we first make only the first leg F 1 of F to act on its eigenvectors H, E, E , u i (generators of U g and of Q), and find F 1 H ⊗ F 2 = H ⊗ 1, where u i ∈ {y i , η i ,∂ i }, 1 ≤ i ≤ 3; note that the exponents ±λ i /2 take the values ±1, 0. By the first relation the twisted tensor or star products are trivial if H or some u 2 is the first factor. The following two imply (153). Moreover, for all u i , v i ∈ {y i , η i ,∂ i }, we find By explicit calculations these imply relations (154-159), as well as (160), once one notes that 2 √ ab ε ijk f i L jk = y 3 E −y 1 E − √ ay 2 H = y 3 E −y 1 E − √ a y 2 H +iνy 1 H −2iν(1+iν)y 1 E.

B.7.2 Metric and principal curvatures on the circular hyperboloids and cone
One can easily check the statements of the first paragraph of section 4.6.1 using e.g. the basis we have abbreviated ρ 2 ≡ (x 1 ) 2 +(x 2 ) 2 . S is orthogonal with respect to g, while S t := {v 1 , v 2 } is an orthogonal basis of Ξ t with respect to g t , since, by an easy computation, where E(x) ≡ f i (x)f i (x) = x i x i . Since E = 2c on M c , M c is indeed Riemannian if c < 0, Lorentzian if c > 0, whereas the metric induced on the cone M 0 is degenerate. One can easily check (161), (162) on such a S t by explicit computations. The dual basis consists of and the same results in the last line if we flip the arguments. To prove (165) we use (67), (56), (177), the X -linearity of g. To prove (163) we use (67), (177). The undeformed version of (164) follows from (162) by X -linearity. We prove (164) using (67), (18), the definition of R: To prove (166) we note that classically ∇ is the Levi-Civita covariant derivative ∇