Publication

$L_{q,p}$-cohomology of Riemannian Manifolds with Negative Curvature

Marc Troyanov
2009
Book Chapters

Abstract

We consider the L_{q,p}-cohomology of a complete simply connected Riemannian manifold (M,g) with pinched negative curvature. The connection between the L_{q,p}-cohomology of (M,g) and Sobolev inequalities for differential forms on (M,g) was established by the authors in a previous publications.

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Marc Troyanov
2009
Book Chapters

Abstract

Official source

Related concepts (23)

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions are smooth functions.

Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

Related publications (4)

$L_{q,p}$-cohomology of Riemannian Manifolds with Negative Curvature

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