Elementary Differential Geometry

Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.

Part 1 Calculus on Euclidean space: Euclidean space; tangent vectors; directional derivatives; curves in R3; 1-forms; differential forms; mappings. Part 2 Frame fields: dot product; curves; the Frenet formulas; arbitrary speed curves; covariant derivatives; frame fields; connection forms; the structural equations. Part 3 Euclidean geometry: isometries of R3; the tangent map of an isometry; orientation; Euclidean geometry; congruence of curves. Part 4 Calculus on a surface: surfaces in R3; patch computations; differentiable functions and tangent vectors; differential forms on a surface; mappings of surfaces; integration of forms; topological properties; manifolds. Part 5 Shape operators: the shape operator of M R3; normal curvature; Gaussian curvature; computational techniques; the implicit case; special curves in a surface; surfaces of revolution. Part 6 Geometry of surfaces in R3: the fundamental equations; form computations; some global theorems; isometries and local isometries; intrinsic geometry of surfaces in R3; orthogonal coordinates; integration and orientation; total curvature; congruence of surfaces. Part 7 Riemannian geometry: geometric surfaces; Gaussian curvature; covariant derivative; geodesics; Clairaut parametrizations; the Gauss-Bonnet theorem; applications of Gauss-Bonnet. Part 8 Global structures of surfaces: length-minimizing properties of geodesics; complete surfaces; curvature and conjugate points; covering surfaces; mappings that preserve inner products; surfaces of constant curvature; theorems of Bonnet and Hadamard.