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A fundamental result of de Rham cohomology is that the kth de Rham cohomology vector space of a manifold M is canonically isomorphic to the Alexander-Spanier cohomology ...

A gadget defined for complex vector bundles. The Chern classes of a complex manifold are the Chern classes of its tangent bundle. The ith Chern class is an obstruction to the ...

A differential ideal I on a manifold M is an ideal in the exterior algebra of differential k-forms on M which is also closed under the exterior derivative d. That is, for any ...

The holomorphic tangent bundle to a complex manifold is given by its complexified tangent vectors which are of type (1,0). In a coordinate chart z=(z_1,...,z_n), the bundle ...

The term metric signature refers to the signature of a metric tensor g=g_(ij) on a smooth manifold M, a tool which quantifies the numbers of positive, zero, and negative ...

If a compact manifold M has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if M has negative curvature, then ...

A point x in a manifold M is said to be nonwandering if, for every open neighborhood U of x, it is true that phi^nU intersection U!=emptyset for a map phi for some n>0. In ...

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 ...

Written in the notation of partial derivatives, the d'Alembertian square ^2 in a flat spacetime is defined by square ^2=del ^2-1/(c^2)(partial^2)/(partialt^2), where c is the ...

If W is a simply connected, compact manifold with a boundary that has two components, M_1 and M_2, such that inclusion of each is a homotopy equivalence, then W is ...