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A continuous real function L(x,y) defined on the tangent bundle T(M) of an n-dimensional smooth manifold M is said to be a Finsler metric if 1. L(x,y) is differentiable at ...

The ith Pontryagin class of a vector bundle is (-1)^i times the ith Chern class of the complexification of the vector bundle. It is also in the 4ith cohomology group of the ...

A section of a solid is the plane figure cut from the solid by passing a plane through it (Kern and Bland 1948, p. 18).

A tubular neighborhood of a submanifold N in M is an embedding of the normal bundle (nu_N) of N into M, i.e., f:nu_N->M, where the image of the zero section of the normal ...

Differential Geometry

Let M^n be a compact n-dimensional oriented Riemannian manifold without boundary, let O be a group representation of pi_1(M) by orthogonal matrices, and let E(O) be the ...

A homogeneous space M is a space with a transitive group action by a Lie group. Because a transitive group action implies that there is only one group orbit, M is isomorphic ...

An orientation on an n-dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an ...

The ith Stiefel-Whitney class of a real vector bundle (or tangent bundle or a real manifold) is in the ith cohomology group of the base space involved. It is an obstruction ...

A vector field is a map f:R^n|->R^n that assigns each x a vector f(x). Several vector fields are illustrated above. A vector field is uniquely specified by giving its ...