' an exciting web-based interface to Gaussian and other computational chemistry packages designed to make is easy to introduce students to electronic structure calculations and to run them from their own personal computers. Highly Recommends WebMO (Gaussian News, December, 2000): Riemannian connections, brackets, proof of the fundamental theorem of Riemannian geometry, induced connection on Riemannian submanifolds, reparameterizations and speed of geodesics, geodesics of the Poincare's upper half plane. Proof of the existence and uniqueness of geodesics.

Integration on manifolds, definition of volume, and proof of the existence of partition of unity.Д¬onnections, covariant derivative, parallel translations, Foucault's pendulum, and geodesics. The notion of distance on a Riemannian manifold and proof of the equivalence of the metric topology of a Riemannian manifold with its original topology. Proof that RP n is oreintable for n odd and is not orientable for n even.Дefinition of a Riemannian metric, and examples of Riemannian manifolds, including quotients of isometry groups and the hyperbolic space. Proof of the nonorientability of the Mobius strip and the nonembeddability of the real projective plane in R 3. Various definitions of orientability and the proof of their equivalence. Proof of Sard's theorem (not yet typeset, but contains some exercises). Regular values, proof of fundamental theorem of algebra, Smooth manifolds with boundary, Sard's theorem, and proof of Brouwer's fixed point theorem. Proof of Whitney's 2n+1 embedding theorem. Proof of the smooth embeddibility of smooth manifolds in Euclidean space. Proofs of the inverse function theorem and the rank theorem. Characterization of tangent space as derivations of the germs of functions. Proof of the embeddibility of comapct manifolds in Euclidean space.Дefinition of differential structures and smooth mappings between manifolds.Дefinition of Tangent space. Proofs of the Cauchy-Schwartz inequality, Heine-Borel and Invariance of Domain Theorems.Дefinition of manifolds and some examples. Review of basics of Euclidean Geometry and Topology. The Gauss-Bonnet theorem, and its applications. Mainardi equations, and Theorema Egregium revisited.Дefinition of geodesic curvature, and the proof that it is intrinsic. Self adjointness of the shape operator, Riemann curvature tensor of surfaces, Gauss and Codazzi The covariant derivative and Lie bracket Riemann curvature tensor and Gauss's formulas Riemann curvature tensor, and a second proof of Gauss's Theorema Egregium. Gauss's formulas, Christoffel symbols, Gauss and Codazzi-Mainardi equations,

Intrinsic metric and isometries of surfaces, Gauss's Theorema Egregium,Д«rioschi's formula for Gaussian curvature. Ratio of areas, and products of principal curvatures. Interpretations of Gaussian curvature as a measure of local convexity, Gaussian curvature, Gauss map, shape operator, coefficients of the firstДЄnd second fundamental forms, curvature of graphs. Torsion, Frenet-Seret frame, helices, spherical curves. The four vertex theorem, Shur's arm lemma, isoperimetric inequality. Osculating circle, Kneser's Nesting Theorem, total curvature, convex curves. General definition of curvature using polygonal approximations (Fox-Milnor's theorem).Д¬urves of constant curvature, the principal normal, signed curvature, turning angle, Isometries of Euclidean space, formulas for curvature of smooth regular curves.

Basics of Euclidean Geometry, Cauchy-Schwarz inequality.Дefinition of curves, examples, reparametrizations, length, Cauchy's integral formula, curves of constant width.

Georgia tech gaussian software