Twisted Quadrics and Algebraic Submanifolds in ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{n}$\end{document}

We propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{n}$\end{document}, specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds ofℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{n}$\end{document}, 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 ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. If we endow ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{n}$\end{document} 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 ⋆-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in ℝ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{3}$\end{document} except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean ℝ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{3}$\end{document} and twisted hyperboloids embedded in twisted Minkowski ℝ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{3}$\end{document} [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].


Introduction
The concept of a submanifold of a manifold plays a fundamental role in mathematics and physics. A metric, connection, ..., on uniquely induces a metric, connection, ..., on . Algebraic submanifolds of affine spaces such as R or C are paramount for their simplicity and their special properties. In the last few decades the program of generalizing differential geometry into so-called Noncommutative Geometry (NCG) has made a remarkable progress [14,35,[41][42][43]; NCG might provide a suitable framework for a theory of quantum spacetime allowing the quantization of gravity (see e.g. [1,20]) or for unifying fundamental interactions (see e.g. [12,15]). 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. [17,45,48,54]). 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 -equivariant noncommutative algebra "of functions on the quantum Euclidean space R ", which is generated by non-commuting coordinates , one can obtain the one A on the quantum Euclidean sphere 1 by imposing that the [central and -invariant] "square distance from the origin" 2 be 1. But the -equivariant differential calculus on A (i.e. the corresponding Abimodule of 1-forms) remains of dimension instead of 1; the 1-form 2 cannot be set to zero, and actually the graded commutator acts as the exterior derivative [11,26,28,53].
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 R , of the Cartan calculus, and of (pseudo)Riemannian geometry on is proposed. In the present work we proceed studying more in detail algebraic submanifolds R , in particular quadrics, using tools of algebraic geometry. Considering C instead of R seems viable, too.
Assume that the algebraic submanifold R consists of solutions of the equations 0 1 2 ...
(1) where 1 ... R R are polynomial functions fulfilling the irreducibility conditions listed in Theorem 1; in particular, the Jacobian matrix is of rank on some non-empty open subset D R , and more precisely consists of the points of D fulfilling (1). One easily shows that E R D is empty or of zero measure 1 . By replacing in (1) , with 1 ... D , we obtain a -parameter family of embedded manifolds ( 0 ) of dimension that are level sets of . Embedded algebraic submanifolds can be obtained by adding more polynomial equations of the same 1 Let be the submatrices of , their determinants, E R 0 , 1 2 ... . E E . At least one polynomial function is not identically zero; hence E has codimension 1 and zero measure, and so has E . type to (1). Let X be the -algebra (over C) of polynomial functions R C, restricted to D . The -algebra X of complex-valued polynomial functions on can be expressed as the quotient of X over the ideal C X of polynomial functions vanishing on : In Appendix A, after recalling some basic notions and notation in algebraic geometry, we prove Theorem 1 Assume that is of rank on a non-empty open subset D R , so that the system (1) defines an algebraic submanifold D of dimension . In addition, assume that is irreducible in C ; this is the case e.g. if there exists a -dimensional affine subspace R meeting in 1 deg points. Then C is the complexification of the ideal generated by the in R 1 ... , i.e. for all C there exist X such that 1 1 .
(In the smooth context, i.e. with D , (3) holds if is of rank on D [32].) X is the quotient of X 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 X the Lie algebra of polynomial vector fields on D (here and below we abbreviate ) and The former is a Lie -subalgebra of , while the latter is a Lie -ideal; both are X --subbimodules. By Theorem 1 the latter decomposes as CC 1 . We identify the Lie algebra of vector fields tangent to with that of derivations of X , namely with A general framework for deforming X into a family -depending on a formal parameter -of noncommutative algebras X over C (the ring of formal power series in with coefficients in C) is Deformation Quantization [6,40]: as a module over C X coincides with X , but the commutative pointwise product of X (C -bilinearly extended to X 1 (6) where are suitable bidifferential operators of degree at most. We wish to deform X into a noncommutative algebra X in the form of a quotient X X C C X (7) with C a two-sided ideal of X , and fulfilling itself X X as an equality of C -modules. To this end we require that C C , i.e. that C for all X , C, so that C for all X and C . 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 X , 1 .. , 0 N (8) (this implies that the 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 g (seen as Hopf algebras) of Lie groups and of their module algebras, based on twisting; a twist is a suitable element (a 2-cocycle, see Section 2.1) (here C , and tensor products are meant completed in the -adic topology); F acts on the tensor product of any two g-modules or module algebras, in particular algebras of functions on any smooth manifolds acts on, including some symplectic manifolds 3 [3]. Given a generic smooth manifold , the authors of [1] pick up g , the Lie algebra of smooth vector fields on (and of the infinite-dimensional Lie group of diffeomorphisms of ), and the -module algebra X ; F 1 F 2 seen as differential operators acting on X have order at most and no zero-order term. The corresponding deformed product reads where F F 1 1 1 1 F 1 F 2 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 X into new Hopf algebra F and F -equivariant module algebra X , but also the -equivariant X -bimodule of differential forms on , 2 In fact, for all 1 C ( X ) (8) implies 1 and, for all X , by the associativity of , 1 1 1 C ; and similarly for . It is not sufficient to require that , belong to C to obtain the same results. 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 1, which do not admit a -product induced by a Drinfel'd twist (c.f. [9,16]). Nevertheless, if one is not taking into account the Poisson structure, every -manifold can be quantized via the above approach. their tensor powers, the Lie derivative, and the geometry on (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 defined by which consists of vector fields tangent to all submanifolds (because they fulfill 0 for all R ) at all points. As in [32], we note that, applying this deformation procedure to the previously defined X with a twist F , we satisfy (8) and therefore obtain a deformation X of X such that for all D X X X C ; moreover, C CC , see Section 2.3. In other words, we obtain a noncommutative deformation, in the sense of deformation quantization and in the form of quotients as in (2) like R , the Heisenberg or the " " group, one can construct even a strict (i.e. non-formal) deformation quantization [52] of such that the -product remains invariant under itself (or a cocommutative Hopf algebra), see e.g. [7,52].
In Section 3 we apply this procedure to algebraic submanifolds R . For simplicity we stick to of codimension 1, and we assume that there is a Lie subalgebra g (of dimension at least 2) of both and the Lie algebra aff of the affine group Aff R R of R ; the level sets of of degree 1 (hyperplanes) or 2 (quadrics) are of this type. Choosing a twist F g g we find that the algebra X of polynomial functions (with complex coefficients) in the set of Cartesian coordinates 1 ... is deformed so that every -polynomial of degree in equals an ordinary polynomial of the same degree in , and vice versa. This implies in particular that the polynomial relations 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 X C can be defined globally in terms of generators and polynomial relations, and moreover the subspaces X , X of X , X X C consisting of polynomials of any degree in coincide as C modules with their deformed counterparts X , X ; in particular their dimensions (hence the Hilbert-Poincaré series of both X and X ) 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 of differential forms, that of differential operators, etc. We convey all these informations into what we name the differential calculus algebras Q Q on R 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 and in the vector fields), and their deformations Q Q (see Sections 3.1 and 3.2). In Section 4 we discuss in detail deformations, induced by unitary twists of 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 of isometries of dimension at least 2; choosing a twist F k k we thus find twisted (pseudo)Riemannian (with the metric given by the twisted first fundamental form) that are symmetric under the Hopf algebra 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 , 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 2 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 2 1 2 ... 1 2 2 2 0 in Minkowski R , are maximally symmetric cosmological solutions to the Einstein equations of general relativity with a nonzero cosmological constant in spacetime dimension 1, and play a prominent role in present cosmology and theoretical physics (see e.g. [19,44]). Interpreting in Minkowski R as relativistic -momentum, rather than position in spacetime, then the same equation represents the dispersion relation of a relativistic particle of square mass 2 . 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 (rather than R ), just dropping -structures and the related constraints on the twist; twist deformations of the (zero-measure) algebraic set E .
Finally, we mention that in [30,31,50] an alternative approach to introduce NC (more precisely, fuzzy) submanifolds R has been proposed and applied to spheres, projecting the algebra of observables of a quantum particle in R , subject to a confining potential with a very sharp minimum on , to the Hilbert subspace with energy below a certain cutoff.
Everywhere we consider vector spaces over the field K R C ; we denote by the K -module of formal power series in with coefficients in K. We shall denote by the same symbol a K-linear map and its K -linear extension .

Hopf Algebras and their Representations
Hopf algebras. We recall that a Hopf algebra over K is an associative unital algebra over K [ is the product: for , K with 1 1 is the unit] endowed with a coproduct, counit, antipode . While are algebra maps, is an anti-algebra map; they have to fulfill a number of properties (see e.g. [13,22,43]), namely id id 2 (coassociativity), id id id (counitality), id id (antipode property). We shall use Sweedler's notation with suppressed summation symbols for the coproduct and its 1 -fold iteration 1 2 ... .
A -involution on a K-algebra A is an involutive, anti-algebra map A A such that for all A and K (here denotes the complex conjugation of ). A Hopf -algebra over K is a Hopf algebra endowed with a -involution such that, for all , The universal enveloping algebra (UEA) g of a K-Lie algebra g is a Hopf algebra; are determined by their actions on 1 and on primitive elements, i.e. g: It is cocommutative, i.e. , where is the flip, . If there is a -involution g g on g such that for all g, the UEA g becomes a Hopf -algebra with respect to the extension g g. Replacing everywhere in the above definition K by the commutative ring K one obtains the definition of a Hopf ( -)algebra over K . For any Hopf ( -)algebra over K the K -linear extension (with completed tensor product in the -adic topology) is trivially a Hopf ( -)algebra over K . Other ones can be obtained by twisting (see below).

Hopf algebra modules and module algebras. Given an associative unital algebra
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 for all 4 . Again, we shall use Sweedler's notation with suppressed summation symbols for the coproduct F and its 1 -fold iteration where R F 21 F is the triangular structure or universal R-matrix. R has inverse R FF 21 R 21 and further satisfies the so-called hexagon relations As the representation theory of a Hopf algebra is monoidal, the K-tensor product M M of two left -modules is also a left -module, via the action 1 2 . The -tensor product is the corresponding monoidal structure on the representation theory of F , since  [24,29]). In other words, , and R are related to F F R by the relations 1 1  is a triangular Hopf -algebra (in fact, R F F R); moreover, A , M are a left F -module -algebra and a F -equivariant A --bimodule when endowed with the undeformed -involutions (cf. [43] Proposition 2.3.7). In particular is a left F -module -algebra. Actually is an isomorphism of the triangular Hopf -algebra onto the one F F , see [1,43] for more information. If F is a unitary twist, then also R is, 1 , and F endowed with the undeformedinvolution is a Hopf -algebra; moreover, A M are respectively a left F -module -algebra and an F -equivariant A --bimodule when endowed with the twisted -involutions (29) where A and M (cf. [29]). In particular is a left F -module -algebra. Actually, one finds that is a triangular Hopfalgebra, in particular , id , and F is an isomorphism of triangular Hopf -algebras, see Proposition 18 in [32].
For their simplicity, here we shall only use abelian or the following Jordanian Drinfel'd twists on UEAs: i.) For a finite number N of pairwise commuting elements 1 1 g we set 1 g g, is a Drinfel'd twist on g ( [51]); it is said of abelian (or Reshetikhin) type. It is unitary if ; this is e.g. the case if the are anti-Hermitian or Hermitian. The twist F exp is both unitary and real, leads to the same R and makes 1, whence F , and the -structure remains undeformed also for --modules and module algebras, see (29). ii.) Let g be elements of a Lie algebra such that 2 . Then defines a Jordanian Drinfel'd twist [49]. If and 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 , where denotes the Lie -algebra of smooth vector fields on a smooth manifold , as a left -module -algebra the -algebra X C of smooth K-valued functions on , as -equivariant symmetric X --bimodules itself, the space of differential 1-forms on , as well as their tensor (or wedge) powers. The Hopf -algebra action on X , and is given by the extension of the Lie derivative: for , X and we have Math Phys Anal Geom (2020) 23: 38 Page 10 of 57 and we set L L L , L 1 id. Henceforth we denote such an extension by .

Twisted Tensor Fields
The tensor algebra T N 0 T on is defined as the direct sum of the K-modules T -times -times (33) for 0, 0, where we set T 0 0 X . Here and below stands for X (rather than K ), namely for all X . Every T is an -equivariant X --bimodule with respect to the module actions Consider a (in particular, unitary or real) Drinfel'd twist F on . Applying the results of Section 2.1 to , X , , and T we obtain the following: F F is a Hopf ( -)algebra, X is a left F -module ( -)algebra, while T are F -equivariant X -( -)bimodules. The F -actions are given by the -Lie derivative L L F 1 F 2 for all F and T . On -vector fields , the -Lie derivative structures as a -Lie algebra. This means that is twisted skew-symmetric, i.e. R 1 R 2 and satisfies the twisted Jacobi identity and -vector fields act on X as twisted derivations, i.e.
for all and X . By setting A T we can apply the results of Section 2.1, in particular define a deformed tensor algebra T with associativetensor product defined by (25). This can be decomposed as In particular, for all T , X and The third formula shows that is actually X , the tensor product over X . Let T . On any local chart of there unique functions 1 Higher order differential forms are defined by the twisted skew-symmetrization of ( -wedge product, an associative unital product), and we define to be the twisted exterior algebra of (see [54] for more information). The dual pairing between vector fields and 1-forms can be equivalently considered as X -bilinear maps X or X ; for all arguments , these maps have the same images, which we respectively denote by the lhs and right-hand side (rhs) of the identity . They have distinct twist deformations ( -pairings) defined by with and respectively. They satisfy for all F , , , or , and As one can extend the ordinary pairing to higher tensor powers setting ...
for all T (the image will belong again to T ) provided or for all , 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 and of the are opposite of each other), properties (42) are preserved, namely the -paring is 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 ).

Twisted Covariant Derivatives and Metrics
Its curvature R F and torsion T F maps respectively act on all through and are left X -linear maps T F and R F fulfilling 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 T T on and its equivariance Lie algebra e (cf. [32]). The latter is a Lie subalgebra of the Lie algebra of vector fields defined by e for all T . (51) It follows that is e-equivariant, i.e. 1 2 for all e, and T . If F e e is a Drinfel'd twist, then defines an e F -equivariant twisted connection F for all , T and (cf. [32] Proposition 2). A metric on is a non-degenerate element g g g such that g g g . We can view g as an element g g g with g g F 1 g F 2 g . 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 (sum over ), where , are -dual bases of , in the sense . One easily finds Ric F R F . For a (pseudo-)Riemannian manifold g we define the Lie subalgebra e. T 0 and L g g g for all ] and e the corresponding equivariance Lie algebra, we obtain k e by the Koszul formula.
The following results are taken from [2,32]. If F k k is a twist "based on Killing vector fields", then (52) defines a twisted LC connection F K T T , and moreover for all . F is the unique LC connection with respect to g ; equivalently for all . This twisted metric map g 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-toone correspondence with an element Ric F such that Ric F Ric F , by the non-degeneracy of the -pairing. The twisted curvature, Ricci tensor and Ricci scalar are 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 D R whose points solve the single equation 0. More generally, the solutions define a smooth manifold ; varying we obtain a whole 1-parameter family of embedded submanifolds R of dimension 1. In [32] X stands for the -algebra of smooth functions on D , and also X C ... are understood in the smooth context.
Twist deformation of tangent and normal vector fields. According to Section 2.1 is a X -bimodule with X -subbimodules C CC . We further define the X -bimodule 0 . (60) By Proposition 9 in [32], the X -bimodules C and C CC are -Lie subalgebras of while CC is a -Lie ideal. Furthermore, we obtain the decomposition X d d X , and the twisted exterior algebras resp. coincide as C -modules with C CC . Let g g g be a (non-degenerate) metric on D with inverse g 1 g 1 g 1 .
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 to D by the same symbols and by k the Lie subalgebra of Killing vector fields with respect to g which are also tangent to D . The deformed analogues of (61) into orthogonal X -bimodules, with respect to g and g 1 respectively. is a -Lie subalgebra of , and are orthogonal with respect to the -pairing and actually 0 . Furthermore, the restrictions are non-degenerate. resp. coincide with as C -modules; and similarly for their -tensor (and -wedge) powers. The orthogonal projections pr , pr , pr and pr and their (unique) extensions to multivector fields and higher rank forms are the C -linear extensions of their classical counterparts. They, as well as , are k F -equivariant. The induced metric (first fundamental form) for the family of submanifolds D , where D , stays undeformed: g F pr pr g pr pr g g .
Defining C C and CC , we further obtain The following proposition assures that every element of can be represented by an element in and every element of can be represented by an element in . Proposition 11 in [32]. For C , C the tangent projections pr , pr respectively belong to and .
Let be the LC connection corresponding to D 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 are F pr F and F pr F K K respectively; the latter yields the curvature R F via (48). We now summarize results of Propositions 3, 12 and 13 in [32]. As g F g , also the twisted second fundamental form, curvature, Ricci tensor and Ricci scalar on are k F -invariant and coincide with the undeformed ones as elements Ric F X , are k F -equivariant maps, and for all they actually reduce to where F 1 F 2 F 3 is the inverse of (18); these maps are left (resp. right) Xlinear in the first (resp. last) argument, 'middle' X -linear otherwise, in the sense g g , etc. Furthermore, the following twisted Gauss equation holds for all The twisted first and second fundamental forms, Levi-Civita connection, curvature tensor, Ricci tensor, Ricci scalar on are finally obtained from the above by applying the further projection X X , which amounts to choosing the 0 manifold out of the family. Of course, one can do the same on any other .  [32]); these relations hold also without . The projection pr (C -linear extension of pr ) on , can be equivalently expressed as

Decompositions (63) in terms of bases or complete sets. In terms of Cartesian
(see Proposition 14 in [32]). By the -bilinearity of g these equations imply in particular in terms of the left and right decompositions , in the bases 1  By the mentioned propositions, every complete set of , e.g. , is also a complete set of ; similarly, every complete set of , e.g. or , is also a complete set of .

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 provided we adopt X Pol R , etc. everywhere. We can assume without loss of generality that the be an irreducible polynomial function 5 . It is interesting to ask for which algebraic submanifolds R the infinite-dimensional Lie algebra admits a nontrivial finite-dimensional subalgebra g over R (or C), so that we can build concrete examples of twisted by choosing a twist F g g of a known type. If are manifestly symmetric under a Lie group 6 K, then such a g exists and contains the Lie algebra k of K (if is maximally symmetric then k is even complete -over X -in ). In general, given any set of vector fields that is complete in 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 themselves close a finite-dimensional Lie algebra g. This means that in (77) const, hence is a quadratic polynomial, and is either a quadric or the union of two hyperplanes (reducible case); moreover g is a Lie subalgebra of the affine Lie algebra aff of R . In the next subsection we find some results valid for all 3 drawing some general consequences from the only assumptions X Pol R and g aff ; in particular, in Sections 3.1, 3.2 we show that the global description of differential geometry on R in terms of generators and relations extends to their twist deformations, in such a way to preserve the spaces consisting of polynomials of any fixed degrees in the coordinates , differential and vector fields chosen as generators. In Section 4 we shall analyze in detail the twisted quadrics embedded in R 3 . 5 If , we find on the second term vanishes and the first is tangent to , as it must be; and similarly on . Having assumed the Jacobian everywhere of maximal rank have empty intersection and can be analyzed separately. Otherwise vanishes on (the singular part of ), so that on the latter a twist built using the will reduce to the identity, and the -product to the pointwise product (see the conclusions). 6 For instance, the sphere 1 is invariant; a cylinder in R 3 is invariant under 2 R; the hyperellipsoid of equation 1  strictly speaking are not encompassed in the above analysis because the Jacobian matrix vanishes at the apex 0 (the only singular point). They are algebraic varieties that are limits of the hyperboloids 0 as 0. If we omit the apex, a cone becomes a disconnected union of two nappes (which are open in R ), and g is spanned not only by the , but also by the central anti-Hermitian element 2 generating dilatations; note that all of them vanish on the apex. Hence g R in this case. If we endow R with the Euclidean metric, the metric matrix is not changed by the above Euclidean changes of coordinates, because the Euclidean group is the isometry group H of R , whereas its nonzero (diagonal) elements are rescaled if we rescale 1 2 . Similarly, if we endow R 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 .

Twisted Differential Calculus on R n by Generators, Relations
Let us abbreviate . We name differential calculus algebra on R the unital associative -algebra Q over C generated by Hermitian elements 1 i The 0 1 play respectively the role of the unit, of Cartesian coordinate functions on R , of differentials of , of partial derivatives with respect to . 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 , of dual frames. The . All are trivially also g-modules; also g is, under the adjoint action. Of course, this aff action is compatible with the relations (83-84); the ideal I generated by their left-hand sides in the free -algebra A generated by 0 1 ... 3 is aff -invariant. The aff -action is also compatible with the invariance of the exterior derivative, because . In the Q framework is the inhomogeneous first order differential operator sum of a first order part (the vector field ) and a zero order part (the multiplication operator by ); it must not be confused with the product of by from the right, which is equal to and so far has been denoted in the same way. In the Q framework we denote the latter by (of course , 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 and the one defined by ( are the total degrees in respectively). Fixing part or all of we obtain the various relevant aff modules or module subalgebras or X -bimodules: .... For instance the exterior algebra is generated by the alone ( 0) and its component is the aff -submodule of exterior -forms ; by (84) 3 dim ; in particular this is zero for , 1 for , and 0 . Let X be the component of X of degree , and X 0 X (i.e. X X consist resp. of homogeneous and inhomogenous polynomials in of degree ); X 0 X is trivially a filtered algebra X 0 X . Let D be the unital subalgebra generated by the alone, D its component of degree , and D 0 D ; then D 0 D is trivially a filtered algebra D 0 D . Finally, let Choosing a twist F based on aff (in particular, on g) and setting (10) for all Q one makes Q into a aff F -module (resp. g F -module) algebra Q with grading (whereas the grading is not preserved). In the appendix we prove  In the Q framework R 1 R 2 , while so far it stood just for the -product of the vector field by the function from the right, i.e. for the first term at the rhs; denoting the latter by R 1 R 2 , we can abbreviate . Of course R 1 R 2 , remain valid. These results are the strict analogues of their untwisted counterparts. Relation (92) is much stronger than the equality of infinite-dimensional C -modules Q Q ; it implies dim Q dim Q over C , so that the Hilbert-Poincaré series of the -graded and -filtered algebras Q Q coincide. In particular, 0 yields dim X dim X . The 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 of the twisted Heisenberg algebra on R (the 0 component of Q ) were already determined in [24,25], while (92) extends results of [29].

Twisted Differential Calculus on M by Generators, Relations
Chosen a basis 1 By (85), (96) 2 the span a (reducible) g--module. Hence A , which is generated by them, is a g-module -algebra, and the A are g--submodules. It is immediate to check that also the span a (reducible) g--module, J more precisely , while more generally is a numerical combination of the appearing in the same equation where appears, e.g. 1 2 . Therefore I is a g--module, and Q is a g-module -algebra as well; moreover I I and Q Q are g -submodules as well. Equations (85) and (103) with a twist F g g imply that:

(this differs from ).
These results are the strict analogue of their undeformed counterparts. Relation (106) is much stronger than the equality of infinite-dimensional C -modules Q Q ; it implies dim Q dim Q over C , so that the Hilbert-Poincaré series of Q Q coincide. In particular, setting 0, we find dim X dim X .
In Section 4 we explictly determine all of the relations 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 transforma-tion. 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 . 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 gbased 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 " " Lie subalgebra of the symmetry Lie algebras. Note that there are other choices of Drinfel'd twists on the " "-Lie algebra. In particular we like to mention the twist of Theorem 2.10 of [34], which is the real (i.e. F 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 calculatecommutators 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 " " 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 of a vector field by a function from the right is understood in the Q Q sense (see Section 3.1) R 1 R 2 . Fig. 1 Overview of the quadrics in R 3 : signs of the coefficients of the equations in canonical form (if not specified, all 00 R are possible), rank, associated symmetry Lie algebra g, type of twist deformation; h 1 stands for the Heisenberg algebra. For fixed each class gives a family of submanifolds parametrized by , except classes (f), (g), (h), which altogether give a single family; so there are 7 families of submanifolds. We can always make 1 1 by a rescaling of . The † reminds that the cone (e) is not a single closed manifold, due to the singularity in the apex; we build an abelian twist for it using also the generator of dilatations Hence the -commutation relations of the g F -equivariant -algebra Q read The -structures on g F , Q Q remain undeformed.
Alternatively, one could twist everything by the unitary abelian twist F exp 12 23 . : a 2 > 0, a 3 = 0, a   The -structures on g F , Q Q remain undeformed. This is essentially the same as Proposition 15 in [32]. Alternatively, as a complete set in instead of 12 13 23 we can use 12 3 , which is actually a basis of ; the Lie algebra g so 2 R generated by the latter is abelian; the relevant relations are (115) 0 ,  [32] . F exp 3 12 is a unitary abelian twist inducing the following twist deformation of g, of Q on R 3 and of Q on the elliptic cylinders (122). The g F counit, coproduct, antipode on 3 12 coincide with the undeformed ones. The twisted star products and Lie brackets of 3 12 coincide with the untwisted ones. The twisted star products of 3 12 with , and those among the latter, equal the untwisted ones, except

Proposition 16 in
In terms of star products 12 1 2 2 1 . Also the relations characterizing the g F -equivariant -algebra Q , i.e. Equation (122), its differential and (76), keep the same form: The -structures on g F , Q Q remain undeformed. The deformation via the abelian twist F exp 3 12 k k yields

Circular Cylinders Embedded in Euclidean
For all fixed 0, R is a foliation of R 3 . The Lie algebra g is spanned by the vector fields 12 To compute the action of F on functions it is convenient to adopt the eigenvectors of as new coordinates. In fact, with 1 2, 2 2 and 3 0. Abbreviating , the inverse coordinate and the partial derivatives transformations read 1 1 2   1  2  1  1  2 1   1  2  1  1  2   2  1  2   2  1  2  1  2 1   1  2  2  2  1   3  3  3  3  3 3 . (135) In the new coordinates The actions of on coordinate functions, differential forms and vector fields are given by for all 1 3 The twisted star products of with equal the untwisted ones, except In terms of star products For every 0, this equation with 0 singles out a variety consisting of two planes intersecting along the -axis; R is a foliation of R 3 . The case 0 is reduced to the case 0 by a 2 rotation around the -axis. Equation (144) can be obtained from the one (131) characterizing the hyperbolic paraboloids (d) setting 0. Hence also the tangent vector fields (or equivalently ), their commutation relations, their actions on the (or equivalently on the defined by (134-135)), the commutation relations of the with the can be obtained from the ones of case (d) by setting 0. The fulfill again (132), or equivalently (133), so that g so 1 1 R 2 . 13 23 is a unitary abelian twist inducing the twisted deformation of g, of Q on R 3 and of Q on the hyperbolic cylinders (144) that is obtained by replacing in Proposition 16 in [32], Section 4.3.

Proposition 8 F exp
We can also deform everything with the same Jordanian twist as in (d). We find

Proposition 9
Setting 0 in Proposition 7 one obtains the deformed g, Q on R 3 and Q on the hyperbolic cylinders (144) induced by the unitary twist F exp 2 log 1 .

Circular Hyperboloids and Cone Embedded in Minkowski R 3
We now focus on the case 1 1 , i.e. . g is equivariant with respect to g, where g so 2 1 is the Lie -algebra spanned by the vector fields , tangent to 1 0 . The first fundamental form g g pr pr makes Riemannian if 0, Lorentzian if 0, whereas is degenerate on the cone 0 . Moreover, where (outward normal); in particular, this implies the proportionality relation 1 2 (here g ) between the matrix elements of g in any basis 1 2 of , and, applying the Gauss theorem, one finds the following components of the curvature and Ricci tensors, Ricci scalar (or Gauss curvature) on : [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 R
]. All diverge as 0 (i.e. in the cone 0 limit). is therefore de Sitter space 2 if 0, the union of two copies of anti-de Sitter space 2  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, (66), (67) apply; namely, on 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 X , ..., 'feel' the twist (compare also to [2] Theorem 7 and eq. 6.138). Also the Ricci scalar (or Gauss curvature) R F remains the undeformed one 1 . By (67) the twisted counterpart of (161) becomes the second equality holds because is k-invariant. Similarly, by (67), (55) for all ; the twisted counterpart of (162) is obtained choosing . Hence the matrix elements of F R F Ric F in any basis are obtained from those of the twisted metric g on . In the appendix we sketchily prove that on Finally, we also show that the twisted Levi-Civita connection on gives We recall that a sheet of the hyperboloid , 0, is equivalent to a hyperbolic plane. Other deformation quantizations of the latter have been done, in particular that of [8] in the framework [7,52] (cf. the introduction). However, while the -product [8] is 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 k F -equivariant i.e. relation (21) holds.

(h) Additional Twist Deformation of the Cone
The equation of the cone 0 in canonical form is (145) with 0. In addition to the tangent vector fields or fulfilling (146) also the generator D of dilatations is tangent to 0 (only), D 0 , since D 2 ; furthermore it commutes with all . Hence the anti-Hermitian elements D span a Lie algebra g so 2 1 R. The actions of on Q are as in cases (e-f), while that of D is determined by (167) Therefore, we can build also abelian twist deformations of 0 of the form F exp D , g. Here we choose 13 2 , i.e. F exp D 2 . The cases with 23 12 are similar. Setting 1 1 3 and 2 0, for we find Having this in mind, in the appendix we easily determine the twist deformed structures.
Proposition 10 F exp D 2 is a unitary abelian twist inducing the following twisted deformation of g, of Q on R 3 and of Q on the cone 0 . The g F counit, coproduct, antipode on D coincide with the undeformed ones, except One may wonder whether the irreducibility in R 1 of each polynomial 1 is a sufficient condition in order that lies in 1 , the ideal generated by 1 . The following example answers in the negative.
Example 11 Consider in R 3 the variety defined by the system  1 2 do, and therefore belongs to the complexification of 1 . As for the last statement, the projective closure P C of the zero locus of (1) in P C has degree at least . On the other hand, is the maximum degree so is a complete intersection in P C . Then, there cannot be other components and the variety defined by (1) obtained from the previous one applyingd

B.2 Proof of Proposition 3
All statements up to (106) and the statement that the -polynomials have the same degrees in as the polynomials are straightforward consequences of (85) and of what precedes the proposition. Under g the transform as the ; in fact, since , we find . Hence   imply (117) 1 . The twisted antipodes (117) 2 follow using the properties F id F id F F . The twisted tensor and star products coincide with the untwisted ones as soon as one of the factors is 13 or 23 . This is because the latter commute with both legs of the twist. Among all star products of generators of g only the one 12 12 0 13 12 23 12 12 12 23 13 is different. By a similar direct calculation one can prove (118). The latter imply (119-120) and that the submanifold constraints coincide with their twisted analogues, namely (121) holds. The twisted star involutions coincide with the untwisted ones, since 13 23 13 23 13 23 0. This concludes the proof of the proposition.

B.5 (c) Family of Elliptic Cylinders
Proof of Proposition 16 in [32], see Section 4.3 Since 3 12 0 and 3 12 are anti-Hermitian, F exp 3 12 is a unitary abelian twist on g. As 3 12 commute with both legs of the twist, and all twisted tensor and star products as well as Lie brackets where one of the factors is 3 12 coincide with the untwisted ones. Furthermore the star products of 12 coincide with the classical ones unless the first leg of the twist acts on 3 . Consequently, (115) 0 implies (124) and the equations (125) coincide with their classical analogues. The twisted star involutions are trivial since 3 12 3 12 3 12 0. This concludes the proof.

Proof of Proposition 7
The anti-Hermitian vector fields and satisfy (133), which implies that F exp 2 log 1 is a unitary Jordanian twist on g. We note that for all 0, which follows by iteratively applying (133). In particular this implies This and F 1 1 F determine F as in (137). For the twisted coproduct of we first remark that 2 2 1 for all 0, which is proven by induction. Then and F 1 1 F, which determine F as in (137). Next, it is straightforward to check that the coproducts F , with , and the antipode property F id F 1 0 determine the twisted antipodes F 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 (generators of g and of Q), and find for all , 1 3; note that the exponents 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 contribute to its action on the , by a direct computation one thus finds the star products (138-140). In particular the twisted tensor and star products are trivial if 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  [32], see Section 4.6

B.7.1 Proof of Proposition 17 in
From the anti-Hermiticity of the vector fields and from 2 it follows that F exp 2 log 1 is a unitary Jordanian twist on g and that the coproducts and antipodes of are exactly as in case (d). Similarly, (176) holds, because it is only based on the relation 2 . Summing the last two equations we find that F is as in (151). The antipode F follows from the antipode property F id F 0. To compute the twisted tensor and star products we first make only the first leg F 1 of F to act on its eigenvectors (generators of g and of Q), and find   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)  . Since and (generators of Q) are all eigenvectors of , choosing as each of them, we immediately find the remaining formulae in (169-170). One finds the involution using the following results: