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A theorem that classifies planar regular closed curves up to regular homotopy by their contour winding numbers (Whitney 1937). In his thesis, S. Smale generalized this result ...

Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form M×V, where M is a four ...

Given a principal bundle pi:A->M, with fiber a Lie group G and base manifold M, and a group representation of G, say phi:G×V->V, then the associated vector bundle is ...

A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional ...

Given a vector space V, its projectivization P(V), sometimes written P(V-0), is the set of equivalence classes x∼lambdax for any lambda!=0 in V-0. For example, complex ...

From the point of view of coordinate charts, the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In ...

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 strong Riemannian metric on a smooth manifold M is a (0,2) tensor field g which is both a strong pseudo-Riemannian metric and positive definite. In a very precise way, the ...

A weak pseudo-Riemannian metric on a smooth manifold M is a (0,2) tensor field g which is symmetric and for which, at each m in M, g_m(v_m,w_m)=0 for all w_m in T_mM implies ...

A weak Riemannian metric on a smooth manifold M is a (0,2) tensor field g which is both a weak pseudo-Riemannian metric and positive definite. In a very precise way, the ...