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Poincaré's lemma says that on a contractible manifold, all closed forms are exact. While d^2=0 implies that all exact forms are closed, it is not always true that all closed ...

If del xF=0 (i.e., F(x) is an irrotational field) in a simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the gradient of a scalar field phi(x), ...

In functional analysis, the term "Poincaré-Friedrichs inequality" is a term used to describe inequalities which are qualitatively similar to the classical Poincaré Inequality ...

For s_1,s_2=+/-1, lim_(epsilon_1->0; epsilon_2->0)1/(x_1-is_1epsilon_1)1/(x_2-is_2epsilon_2) =[PV(1/(x_1))+ipis_1delta(x_1)][PV(1/(x_2))+ipis_2delta(x_2)] ...

The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk {x in R^2:|x|<1}, with hyperbolic metric ...

Let G denote the group of germs of holomorphic diffeomorphisms of (C,0). Then if |lambda|!=1, then G_lambda is a conjugacy class, i.e., all f in G_lambda are linearizable.

Let {y^k} be a set of orthonormal vectors with k=1, 2, ..., K, such that the inner product (y^k,y^k)=1. Then set x=sum_(k=1)^Ku_ky^k (1) so that for any square matrix A for ...

Solutions to holomorphic differential equations are themselves holomorphic functions of time, initial conditions, and parameters.

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 ...

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds. Before stating Thurston's geometrization conjecture in ...