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...

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

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

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

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...

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

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

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...

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

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...

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

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...

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

The fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection...

vector spaces). This results in a Riemannian manifold. Generally, straight lines do not exist in a Riemannian manifold, but their role is played by geodesics...

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

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, there...

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

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...

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...

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

\beta +(-1)^{k}\alpha \wedge d\beta .} On a Riemannian manifold, or more generally a pseudo-Riemannian manifold, the metric defines a fibre-wise isomorphism...

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...

decomposition theorem, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible...

In differential geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G...

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...

Let ( M , g ) {\displaystyle (M,g)} be a Riemannian manifold, and S ⊂ M {\displaystyle S\subset M} a Riemannian submanifold. Define, for a given p ∈ S {\displaystyle...

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

Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class)...