Riemannian Manifolds | Gaoming Wang

Definition 1.1.1

A Riemannian manifold is a smooth manifold \(M\) equipped with a Riemannian metric \(g\), which is a smooth, positive-definite symmetric \((0,2)\)-tensor field.

We use either \(g(X,Y)\) or \(\left< X,Y \right>\) to denote the inner product of two tangent vectors \(X, Y \in TM\) with respect to the metric \(g\).

The Riemann tensor \(R=R_M\) is defined by

\[ R(X,Y)Z =-\nabla_X \nabla_Y Z + \nabla_Y \nabla_X Z + \nabla_{[X,Y]} Z, \]

Here, \(\nabla\) is the Levi-Civita connection associated with \(g\), i.e., it satisfies the following properties:

Torsion-free: \(\nabla_X Y - \nabla_Y X = [X,Y]\) for all vector fields \(X, Y\).
Metric compatibility: \(X(g(Y,Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)\) for all vector fields \(X, Y, Z\).

The convention for the \((0,4)\)-type Riemann curvature tensor is

\[ R(X,Y,Z,W)=g(R(X,Y)Z,W). \]

If the ambient manifold is Euclidean space \(\mathbb{R}^N\) with the standard metric, we write \(X\cdot Y\) for the inner product of \(X\) and \(Y\), and \(D\) for the standard connection.

Gaoming Wang

Assistant Professor