1. The Equations for Hypersurfaces
Covariant differentiation in a submanifold of a Riemannian manifold.
The second fundamental form, the Gauss formulas, and Gauss' equation;
Synge's inequality. The Weingarten equations and the Codazzi-Mainardi
equations for hypersurfaces. The classical tensor analysis description.
The moving frame description.
Addendum. Auto-parallel and totally geodesic submanifolds.
2. Elements of the Theory of Surfaces in Euclidean 3-Space
The first and second fundamental forms. Classification of points on a surface;
the osculating paraboloid and the Dubin indicatrix. Principal directions
and curvatures, asymptotic directions, flat points and umbilics; all-umbilic
surfaces. The classical Gauss formulas, Weingarten equations, Gauss
equation, and Codazzi-Mainardi equations. Fundamental theorem of
surface theory. The third fundamental form. Convex surfaces; Hadamard's
theorem. The fundamental equations via moving frames. Review of
Lie groups. Applications of Lie groups to surface theory; the fundamental
equations and the structural equations of SO(3). Affine surface theory;
the osculating paraboloids and the affine invariant conformal structure.
The special affine first fundamental form. Quadratic and cubic forms;
apolarity. The affine normal direction; the special affine normal.
The special affine Gauss formulas and special affine second fundamental
form. The Pick invariant; surfaces with Pick invariant 0. The special
affine Weingarten formulas. The special affine Codazzi-Mainardi equations;
the fundamental theorem of special affine surface theory.
3. A Compendium of Surfaces
Basic calculations. The classical flat surfaces. Ruled surfaces. Quadric
surfaces. Surfaces of revolution; rotation surfaces of constant curvature.
Addendum. Envelopes of 1-parameter families of planes.
4. Curves on Surfaces
Normal and geodesic curvature. The Darboux frame; geodesic torsion.
Laguerre's theorem. General properties of lines of curvature, asymptotic
curves, and geodesics. The Beltrami-Enneper theorem. Lines of curvature
and Dupin's theorem. Conformal maps of Euclidean 3-space; Liouville's
theorem. Geodesics and Clairaut's theorem. Special parameter curves.
Singularities of line fields.
5. Complete Surfaces of Constant Curvature
Hilbert's lemma; complete surfaces of constant curvature K>0.
Analysis of flat surfaces; the classical classification of developable surfaces.
Complete flat surfaces. Complete surfaces of constant curvature K<0.
6. The Gauss-Bonnet Theorem and Related Topics
The connection form for an orthonormal moving frame on a surface;
the change in angle under parallel translation. The integral of K dA
over a polygonal region. The Gauss-Bonnet theorem; consequences.
Total absolute curvature of surfaces. Surfaces of minimal total absolute
curvature. Total curvature of curves; Fenchel's theorem, and the
Addenda. Compact surfaces with constant negative curvature.
The degree of the normal map.