7. Higher Dimensions and Codimensions
A. THE GEOMETRY OF CONSTANT CURVATURE MANIFOLDS
The standard models of the spheres and hyperbolic spaces.
Stereographic projection and the conformal model of hyperbolic space.
Conformal maps of Euclidean n-space and the isometries of
hyperbolic space. Totally geodesic submanifolds and geodesic spheres
of hyperbolic space. Horospheres and equidistant hypersurfaces.
Geodesic mappings; the projective model of hyperbolic space;
B. CURVES IN A RIEMANNIAN MANIFOLD
Frenet frames and curvatures. Curves whose jth curvature vanishes.
C. THE FUNDAMENTAL EQUATIONS FOR SUBMANIFOLDS
The normal connection and the Weingarten equations. Second
fundamental forms and normal fundamental forms; the Codazzi-Mainardi
equations. The Ricci equations. The fundamental theorem for submanifolds
of Euclidean space. The fundamental theorem for submanifolds of constant
D. FIRST CONSEQUENCES
The curvatures of a hypersurface; Theorema Egregium; formula for the
Gaussian curvature. The mean curvature normal; umbilics; all-umbilic
submanifolds of Euclidean space. All-umbilic submanifolds of constant
curvature manifolds. Positive curvature and convexity.
E. FURTHER RESULTS
Flat ruled surfaces in Euclidean space. Flat ruled surfaces in constant
curvature manifolds. Curves on hypersurfaces.
F. COMPLETE SURFACES OF CONSTANT CURVATURE
Modifications of results for surfaces in Euclidean 3-space. Surfaces of|
constant curvature in the 3-sphere: surfaces with constant curvature 0;
the Hopf map. Surfaces of constant curvature in hyperbolic 3-space:
Jörgens theorem; surfaces of constant curvature 0; surfaces of constant
curvature -1; rotation surfaces of constant curvature between -1 and 0.
G. HYPERSURFACES OF CONSTANT CURVATURE IN HIGHER DIMENSIONS
Hypersurfaces of constant curvature in dimensions >3. The Ricci tensor;
Einstein spaces, hypersurfaces which are Einstein spaces. Hypersurfaces
of the same constant curvature as the ambient manifold.
Addenda. The Laplacian.
The * operator and the Laplacian on forms; Hodge's theorem.
When are two Riemannian manifolds isometric? Better imbedding invariants.
8. The Second Variation
Two-parameter variations; the second variation formula. Jacobi fields;
conjugate points. Minimizing and non-minimizing geodesics.
The Hadamard-Cartan Theorem. The Sturm Comparison Theorem;
Bonnet's Theorem. Generalizations to higher dimensions; the
Morse-Schoenberg Comparison Theorem; Meyer's Theorem; the
Rauch Comparison Theorem. Synge's lemma; Synge's Theorem.
Cut points; Klingenberg's theorem.
9. Variations of Length, Area, and Volume
Variation of are for normal variations of surfaces in Euclidean 3-space;
minimal surfaces. Isothermal coordinates on minimal surfaces:
Bernstein's Theorem. Weierstrass-Enneper representation. Associated
minimal surfaces; Schwarz's Theorem. Change of orientation;
Henneberg's minimal surface. Classical calculus of variations in n dimensions.
Variation of volume formula. Isoperimetric problems. Isothermal coordinates.
Immersed spheres with constant mean curvature. Imbedded surfaces with
constant mean curvature. The second variation of volume.