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An integer kappa equal to 0 or 1 which vanishes iff the product manifold M^4×R can be given a smooth structure. Here, M^n is a compact connected topological four-manifold.

A generalization of calculus of variations which draws the relationship between the stationary points of a smooth real-valued function on a manifold and the global topology ...

A continuous vector bundle is a vector bundle pi:E->M with only the structure of a topological manifold. The map pi is continuous. It has no smooth structure or bundle metric.

A homotopy from one embedding of a manifold M in N to another such that at every time, it is an embedding. The notion of isotopy is category independent, so notions of ...

Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions psi_i, such that the support ...

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 ...

A differential ideal is an ideal I in the ring of smooth forms on a manifold M. That is, it is closed under addition, scalar multiplication, and wedge product with an ...

A symplectic form on a smooth manifold M is a smooth closed 2-form omega on M which is nondegenerate such that at every point m, the alternating bilinear form omega_m on the ...

A theorem which states that the analytic and topological "indices" are equal for any elliptic differential operator on an n-dimensional compact smooth C^infty boundaryless ...

Over a small neighborhood U of a manifold, a vector bundle is spanned by the local sections defined on U. For example, in a coordinate chart U with coordinates (x_1,...,x_n), ...