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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 ...

Another word for a C^infty (infinitely differentiable) manifold, also called a differentiable manifold. A smooth manifold is a topological manifold together with its ...

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 ...

The motivating force of topology, consisting of the study of smooth (differentiable) manifolds. Differential topology deals with nonmetrical notions of manifolds, while ...

Donaldson (1983) showed there exists an exotic smooth differential structure on R^4. Donaldson's result has been extended to there being precisely a continuum of ...

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.

Given any open set U in R^n with compact closure K=U^_, there exist smooth functions which are identically one on U and vanish arbitrarily close to U. One way to express this ...

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), ...

Graph smoothing, also known as smoothing away or smoothing out, is the process of replacing edges e^'=v_iv_j and e^('')=v_jv_k incident at a vertex v_j of vertex degree 2 by ...

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 ...