Around 1923, Élie Cartan introduced affine connections on manifolds and defined the main related concepts: torsion, curvature, holonomy groups. He discussed applications of these concepts in Classical and Relativistic Mechanics; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in Galilean Mechanics and, more generally, can...

For an integral homology 3-sphere embedded asymptotically flatly in an Euclidean space, we find a natural framing extending the standard trivialization on the asymptotically flat part. Suppose M is a 3-dimensional closed smooth manifold which has the same integral homology groups as the 3-sphere S. x0 is a fixed point in M . Embed M in a Euclidean space R such that x0 is the infinite point of t...

Let M be a complex projective Fano manifold whose Picard group is isomorphic to Z and the tangent bundle TM is semistable. Let Z ⊂ M be a smooth hypersurface of degree strictly greater than degree(TM)(dimC Z−1)/(2 dimC Z−1) and satisfying the condition that the inclusion of Z in M gives an isomorphism of Picard groups. We prove that the tangent bundle of Z is stable. A similar result is proved ...

We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on PicC which are linearly equivalent to 2Θ. The embedded tangent space at a semi-stable non-stable bundle ξ ⊕ ξ, where ξ is a degree zero line bundle, is shown to consist of those divisors in |2Θ| which contain Sing(Θξ) where Θξ is the translate of Θ by...

Frames for R can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word “frame” has a different meaning in the context of differential geometry and topology. A moving frame for the tangent bundle of a smooth manifold is a basis for the tangent space at each point whi...

We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite–dimensional, compact, classical phase spaces C, by explicitly constructing Hilbert–space vector bundles over C. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic t...

We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More specifically, we consider reparametrization invariant metrics G on the space Imm(S,R) of parametrized regular curves. For many metrics the tangent space TcImm(S ,R) at each curve c splits into vertical and horizontal components (with respect to the projection onto the shape space Bi(S ,R...

In this section we will define the sheaf of relative differential forms of one scheme over another. In the case of a nonsingular variety over C, which is like a complex manifold, the sheaf of differential forms is essentially the same as the dual of the tangent bundle in differential geometry. However, in abstract algebraic geometry, we will define the sheaf of differentials first, by a purely ...

A generalized almost tangent structure on the big tangent bundle T M associated to an almost tangent structure on M is considered and several features of it are studied with a special view towards integrability. Deformation under a βor a B-field transformation and the compatibility with a class of generalized Riemannian metrics are discussed. Also, a notion of tangentomorphism is introduced as ...

The Lie derivation of multivector fields along multivector fields has been introduced by Schouten (see cite{Sc, S}), and studdied for example in cite{M} and cite{I}. In the present paper we define the Lie derivation of differential forms along multivector fields, and we extend this concept to covariant derivation on tangent bundles and vector bundles, and find natural relations between them and...