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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 ...

A type of flow technically defined in terms of the tangent bundle of a manifold.

An ambient isotopy from an embedding of a manifold M in N to another is a homotopy of self diffeomorphisms (or isomorphisms, or piecewise-linear transformations, etc.) of N, ...

The Chern number is defined in terms of the Chern class of a manifold as follows. For any collection Chern classes such that their cup product has the same dimension as the ...

For a smooth harmonic map u:M->N, where del is the gradient, Ric is the Ricci curvature tensor, and Riem is the Riemann tensor.

A closed embedded smooth plane curve has at least four vertices, where a vertex is defined as an extremum of curvature.

A smooth curve which corresponds to the limiting case of a histogram computed for a frequency distribution of a continuous distribution as the number of data points becomes ...

An ordinary knot in three dimensions suspended in four dimensions to create a knotted 2-sphere. Suspended knots are not smooth at the poles.

An anchor is the bundle map rho from a vector bundle A to the tangent bundle TB satisfying 1. [rho(X),rho(Y)]=rho([X,Y]) and 2. [X,phiY]=phi[X,Y]+(rho(X)·phi)Y, where X and Y ...

A Fréchet space is a complete and metrizable space, sometimes also with the restriction that the space be locally convex. The topology of a Fréchet space is defined by a ...