Riemannian manifold

This article defines a differential manifold with the following additional structure -- the structure group is reduced to: orthogonal group

Definition

A Riemannian manifold is the following data:

1. A differential manifold $M$
2. A Riemannian metric $g$ on $M$, viz a smooth association of a positive definite symmetric bilinear form $g_p$ for the tangent space $T_p(M)$.

Relation with other structures

Stronger structures

• Oriented Riemannian manifold

Weaker structures

• Pseudo-Riemannian manifold
• Metric space Further information: Riemannian manifold is metric space