Applications to Classical Geometries
Let X be a manifold with a Riemannian metric g. For any submanifold S ⊂ X, the restriction of the metric generates an odd volume form dgS defined on S, which is called Riemannian volume form. For any bounded function f on X with compact support and any closed submanifold S, the integral
Rgf (S) =
will be called metric integral transform of f . A submanifold S of X is called totally geodesic if it is connected, and any geodesic Γ in X tangent to S lies entirely in S. Any complete, simply connected Riemannian manifold X of constant sectional curvature has many geodesic submanifolds. For any point x ∈ X and any subspace V ⊂ Tx (X) , there is a (unique) geodesic submanifold S through x, such that Tx (S) = V.