Geodesic Curvature - Some Results Involving Geodesic Curvature

Some Results Involving Geodesic Curvature

• The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold . It does not depend on the way the submanifold sits in .

• On the contrary the normal curvature depends strongly on how the submanifold lies in the ambient space, but marginally on the curve: only depends on the point on the submanifold and the direction, but not on .

• In general Riemannian geometry the derivative will be computed using the Levi-Civita connection of the ambient manifold. It splits into a tangent part and a normal part to the submanifold:, and the tangent part is then the usual derivative in (see Gauss equation in the Gauss-Codazzi equations).

• The Gauss–Bonnet theorem.