RiemannianMetric - crowlogic/arb4j GitHub Wiki

A Riemannian metric $g$ on a smooth manifold $M$ is a smooth assignment of an inner product $g_p$ on the tangent space $T_pM$ at each point $p\in M$, varying smoothly with $p$.

More formally, $g$ is a smooth $(0,2)$-tensor field on $M$ that is symmetric and positive-definite at each point $p\in M$. That is, for any smooth vector fields $X,Y,Z$ on $M$, $g$ satisfies the following properties:

• $g(X,Y)=g(Y,X)$ (symmetry)
• $g(X,X) \geq 0$ and $g(X,X) = 0$ if and only if $X=0$ (positive-definiteness)
• $g(X,Y+Z) = g(X,Y) + g(X,Z)$ (linearity)

This allows us to define lengths of curves and angles between tangent vectors, as well as notions of curvature and geodesics on the manifold.

The Riemannian metric $g$ induces a Riemannian distance function $d$ on $M$, given by

$$d (p, q) = \inf { \int_a^b \sqrt{g (\dot{\gamma} (t), \dot{\gamma}(t))} d t : \gamma \text{ is a smooth curve from p to q } }$$

The infimum is taken over all smooth curves $\gamma:[a,b]\to M$ with $\gamma(a) = p$ and $\gamma(b) = q$.

The Riemannian metric also induces a Levi-Civita connection $\nabla$, which is uniquely determined by the properties that it is metric-compatible (i.e., $\nabla g = 0$) and torsion-free.

In summary, a Riemannian metric provides a way to measure distances, angles, and curvature on a smooth manifold, and is a fundamental object in Riemannian geometry.