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Every smooth manifold M has a tangent bundle TM, which consists of the tangent space TM_p at all points p in M. Since a tangent space TM_p is the set of all tangent vectors ...

Every closed three-manifold with finite fundamental group has a metric of constant positive scalar curvature, and hence is homeomorphic to a quotient S^3/Gamma, where Gamma ...

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 ...

A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse.

Distinguish between smooth manifolds in four dimensions.

An h-cobordism is a bordism W between two manifolds M_1 and M_2 such that W is simply connected and the inclusion maps M_1->W and M_2->W are homotopy equivalences.

Consider the set of compact n-Riemannian manifolds M with diameter(M)<=d, Volume(M)>=V, and |K|<=kappa where kappa is the sectional curvature. Then there is a bound on the ...

Every compact 3-manifold is the connected sum of a unique collection of prime 3-manifolds.

Two manifolds are said to be diffeomorphic if there exists a diffeomorphism between them.

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