Riemannian Geometry

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Riemannian geometry is the study of Riemannian manifolds, smooth manifolds that are equipped with a positive definite symmetric rank (2,0) tensor field g, called the metric.

Riemannian Distance

We can turn a Riemannian manifold into a metric space by defining the distance between points p and q to be

$LaTeX: \min_\gamma \int_\gamma ds = \int_0^1 \sqrt{g(\gamma'(t),\gamma'(t))} dt,$

where $LaTeX: \gamma:(0,1) \to M$ is a curve connecting p and q.

To see that this coincides in the Euclidean case, assume first we have two points p and q in $LaTeX: R^n$ that lie along the $LaTeX: x^1$ axis. Then for any curve $LaTeX: \gamma = (\gamma^i)$ connecting them, we have

$LaTeX: \int_0^1 \sqrt{g(\gamma'(t), \gamma'(t))} dt = \int_0^1 \sqrt{\sum_i {\gamma^i}'(t)^2} dt \ge \int_0^1 |{\gamma^i}'(t)| dt \ge \int_0^1 {\gamma^i}'(t) dt = \gamma(1) - \gamma(0) = q - p.$

Suppose now that p and q are arbitrary points in $LaTeX: R^n$ . We can find an element A of the Euclidean group mapping p and q to the $LaTeX: x^1$ axis. Since A preserves the metric, for any curve $LaTeX: \gamma$ connecting p and q we have that

$LaTeX: \int_0^1 \sqrt{g(\gamma'(t), \gamma'(t))} dt = \int_0^1 \sqrt{g(A_* \gamma'(t), A_* \gamma'(t))} dt = \int_0^1 \sqrt{g((A\circ\gamma)'(t), (A\circ\gamma)'(t))} dt.$

Since $LaTeX: A \circ \gamma$ is a curve connecting two points on the $LaTeX: x^1$ axis, we know that the smallest value of the above integral is the Euclidean distance between them, which is the Euclidean distance between p and q.

Riemannian Geometry

From Spinorial wiki

Riemannian Distance

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