Search results for "Tangent space"
showing 10 items of 16 documents
Embeddings of Danielewski surfaces
2003
A Danielewski surface is defined by a polynomial of the form P=x nz −p(y). Define also the polynomial P ′ =x nz −r(x)p(y) where r(x) is a non-constant polynomial of degree ≤n−1 and r(0)=1. We show that, when n≥2 and deg p(y)≥2, the general fibers of P and P ′ are not isomorphic as algebraic surfaces, but that the zero fibers are isomorphic. Consequently, for every non-special Danielewski surface S, there exist non-equivalent algebraic embeddings of S in ℂ3. Using different methods, we also give non-equivalent embeddings of the surfaces xz=(y d n >−1) for an infinite sequence of integers d n . We then consider a certain algebraic action of the orthogonal group $\mathcal O(2)$ on ℂ4 which was…
Affine varieties and lie algebras of vector fields
1993
In this article, we associate to affine algebraic or local analytic varieties their tangent algebra. This is the Lie algebra of all vector fields on the ambient space which are tangent to the variety. Properties of the relation between varieties and tangent algebras are studied. Being the tangent algebra of some variety is shown to be equivalent to a purely Lie algebra theoretic property of subalgebras of the Lie algebra of all vector fields on the ambient space. This allows to prove that the isomorphism type of the variety is determinde by its tangent algebra.
Curves with no tritangent planes in space and their convex envelopes
1990
M. H. Freedman ([3]) proved that for a generic subset of closed curves in ~ 3 with nonvanishing curvature and torsion the number of t r i tangent planes is even and finite. He also guessed, for each even number s _> 0, the existence of an open subset A8 of closed curves with nonvanishing curvature and torsion such tha t each curve in A8 has exact ly s t r i t angent planes. A question tha t can be asked in this context is: Which curves with nonvanishing curvature and torsion have no t r i tangent planes? An example of such a curve is given by the (1,2)-curve on the torus with rat io a, 3 < a < 5 (see [2]). For a generi c curve, we give a pa r t i a l answer to this question here by finding …
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below
2013
We show that in any infinitesimally Hilbertian CD* (K,N)-space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian CD* (0,N)-spaces.
Covariant phase space quantization of the Jackiw-Teitelboim model of two-dimensional gravity
1992
Abstract On the basis of the covariant phase space formulation of field theory we analyze the Jackiw-Teitelboim model of two-dimensional gravity on a cylinder. We compute explicitly the symplectic structure showing that the (reduced) phase space is the cotangent bundle of the space of conjugacy classes of the PSL(2, R ) group. This makes it possible to quantize the theory exactly. The Hilbert space is given by the character functions of the PSL (2, R ) group. As a byproduct, this implies the complete equivalence with the PSL (2, R )-topological gravity model.
Covariant phase-space quantization of the induced 2D gravity
1993
Abstract We study in a parallel way the induced 2D gravity and the Jackiw-Teitelboimmodel on the cylinder from the viewpoint of the covariant description of canonical formalism. We compute explicity thhe symplectic structure of both theories showing that their (reduced) phase spaces are finite-dimensional cotangent bundles. For the Jackiw-Teitelboim model the base space (configuration space) is the space of conjugacy classes of the PSL(2, R ) group. For the induced 2D gravity, and Λ > 0, the (reduced) phase space consist of two (identical) connected components each one isomorphic to the contangent bundle of the space of hyperbolic conjugacy classes of the PSL (2, R ) group, whereas for Λ R …
Geometric Aspects of Mechanics
2010
In many respects, mechanics carries geometrical structures. This could be felt very clearly at various places in the first four chapters. The most important examples are the structures of the space–time continua that support the dynamics of nonrelativistic and relativistic mechanics, respectively. The formulation of Lagrangian mechanics over the space of generalized coordinates and their time derivatives, as well as of Hamilton–Jacobi canonical mechanics over the phase space, reveals strong geometrical features of these manifolds.
Triangular Bézier Approximations to Constant Mean Curvature Surfaces
2008
We give a method to generate polynomial approximations to constant mean curvature surfaces with prescribed boundary. We address this problem by finding triangular Bezier extremals of the CMC-functional among all polynomial surfaces with a prescribed boundary. Moreover, we analyze the $\mathcal{C}^1$ problem, we give a procedure to obtain solutions once the tangent planes for the boundary curves are also given.
Left-Right Equivalence and Stability
2020
We introduce the key equivalence relations on germs of maps, which play an important role throughout the book—right-equivalence and left-right equivalence (A-equivalence). These are induced by groups of diffeomorphisms, so equivalence classes have tangent spaces, and we calculate many examples, including some multi-germs. We introduce the notions of stability and finite determinacy, and prove Mather’s infinitesimal criterion for stability.
Tangent lines and Lipschitz differentiability spaces
2015
We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space contains at least $n$ distinct tangent lines, obtained as the blow-up of $n$ Lipschitz curves, whe…