Pseudo-Riemannian Manifolds# Pseudo-Riemannian Manifolds Euclidean Spaces Euclidean Spaces Spheres smoothly embedded in Euclidean Space Operators for vector calculus Pseudo-Riemannian Metrics and Degenerate Metrics Levi-Civita Connections Pseudo-Riemannian submanifolds Degenerate Metric Manifolds Degenerate manifolds Degenerate submanifolds