In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature...

mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere...

specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described...

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an...

sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold...

allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in...

of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of...

differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic...

geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly...

Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics...

in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization...

Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products. Every Finsler manifold becomes...

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to...

In differential geometry, a hyperkähler manifold is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} endowed with three integrable almost complex...

geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the...

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends...

associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces...

metrics. A pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of the notion of Riemannian manifold where the inner...

of the kinetic term. If one considers a Riemannian manifold or a pseudo-Riemannian manifold, the Riemannian metric induces a linear isomorphism between...

global Riemannian geometry. It goes back to questions of Heinz Hopf from 1931. A modern formulation is: A compact, even-dimensional Riemannian manifold with...

List of formulas in Riemannian geometry Christoffel symbols Intrinsic metric Pseudo-Riemannian manifold Sub-Riemannian manifold Finsler geometry General...

Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian...

of the definitions given below. Connection Curvature Metric space Riemannian manifold See also: Glossary of general topology Glossary of differential geometry...

In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion...

embedding Riemannian manifolds in Euclidean space. The core of his proof was a novel "small perturbation" result, showing that if a Riemannian metric could...

geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the...

the manifold; a manifold with a (positive-definite) metric is a Riemannian manifold, one with an indefinite metric is a pseudo-Riemannian manifold. Thus...

normal to the surface. More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form F(P) = 0 such that dF is...

three-dimensional Riemannian manifolds and four-dimensional Lorentzian manifolds. Schoen and Yau established an induction on dimension by constructing Riemannian metrics...

mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally...