nLab Riemannian manifold

Definition

A Riemannian manifold is a smooth manifold equipped with a Riemannian metric.

See also pseudo-Riemannian manifold

Properties

As $\mathbb{R}$-Riemannian manifolds

$\;$normed division algebra$\;$	$\;\mathbb{A}\;$	$\;$Riemannian $\mathbb{A}$-manifolds$\;$	$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$	$\;\mathbb{R}\;$	$\;$Riemannian manifold$\;$	$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$	$\;\mathbb{C}\;$	$\;$Kähler manifold$\;$	$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$	$\;\mathbb{H}\;$	$\;$quaternion-Kähler manifold$\;$	$\;$hyperkähler manifold$\;$
$\;$octonions$\;$	$\;\mathbb{O}\;$	$\;$Spin(7)-manifold$\;$	$\;$G₂-manifold$\;$

(Leung 02)

Examples

• Euclidean space

• round sphere, Riemann sphere

• Taub-NUT space, ALE space

Related concepts

• To be distinguished from Riemann surface. In particular, the concepts of 2-dimensional Riemannian manifolds and Riemann surfaces are, while closely related, crucially different.

• There is a refinement of topological cobordism categories to one of Riemannian cobordisms.

• See also the Myers–Steenrod theorem.

• For infinite-dimensional manifolds see also orthogonal structure.

• geodesic completeness

• totally geodesic submanifold

• The analog in complex geometry is the notion of Kähler manifold.

• vielbein field, Riemannian metric, affine connection,

• Ricci tensor, scalar curvature

• Riemannian orbifold

  • conical singularity

• hyperbolic manifold

• analytic torsion

• Lorentzian manifold, causal manifold

• Nash embedding theorem

References

• John M. Lee, Riemannian manifolds. An introduction to curvature. Graduate Texts in Mathematics 176 (1997), Springer. ISBN: 0-387-98271-X.

  Second Edition (retitled): Introduction to Riemannian Manifolds (2018), Springer. ISBN: 978-3-319-91754-2 (doi:10.1007/978-3-319-91755-9)

Exposition with an eye towards application in mathematical physics:

• Tohru Eguchi, Peter Gilkey, Andrew Hanson, Section 3 of: Gravitation, gauge theories and differential geometry, Physics Reports 66 6 (1980) 213-393 [doi:10.1016/0370-1573(80)90130-1]