This is a property of a two dimensional Riemannian manifold.

The spherical mean is also defined for Riemannian manifolds in a natural manner.

Let be a compact Riemannian manifold of even dimension .

As in the case of Riemannian manifolds, let .

Subharmonic functions can be defined on an arbitrary Riemannian manifold.

The manifold then has the structure of a 2-dimensional Riemannian manifold.

They admit generalizations to surfaces embedded in more general Riemannian manifolds.

Any harmonic function on a compact connected Riemannian manifold is a constant.

Riemannian manifolds with constant sectional curvature are the most simple.

Variations on the above example include the orthonormal frame bundle of a Riemannian manifold.

A symmetric space is a Riemannian manifold with an isometric reflection across each point.

They are the only complete, simply connected Riemannian manifolds of given sectional curvature.

In dealing with a vector field on a semi Riemannian manifold (p.ex.

A space form is by definition a Riemannian manifold with constant sectional curvature.

Given two Riemannian manifolds of the same dimension, it is not always obvious whether they are locally isometric.

Riemannian manifolds with special holonomy play an important role in string theory compactifications.

These isospectral Riemannian manifolds have the same local geometry but different topology.

Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered.

If is a compact Riemannian manifold, then each equivalence class in contains exactly one harmonic form.

Suppose is a compact two-dimensional Riemannian manifold with boundary .

Suppose first that M is an oriented Riemannian manifold.

A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry.

For any d-dimensional Riemannian manifold the equilateral dimension is at least d + 1.

Let be smooth Riemannian manifolds of respective dimensions .

This generalizes the notion of geodesic for Riemannian manifolds.