Differential Geometry

Budapest Semesters in Mathematics

Lecture Notes by Balázs Csikós


CONTENTS


Unit 1. Basic Structures on Rn, Length of Curves.

Addition of vectors and multiplication by scalars, vector spaces over R, linear combinations, linear independence, basis, dimension, linear and affine linear subspaces, tangent space at a point, tangent bundle; dot product, length of vectors, the standard metric on Rn; balls, open subsets, the standard topology on Rn, continuous maps and homeomorphisms; simple arcs and parameterized continuous curves, reparameterization, length of curves, integral formula for differentiable curves, parameterization by arc length.


Unit 2. Curvatures of a Curve

Convergence of k-planes, the osculating k-plane, curves of general type in Rn, the osculating flag, vector fields, moving frames and Frenet frames along a curve, orientation of a vector space, the standard orientation of Rn, the distinguished Frenet frame, Gram-Schmidt orthogonalization process, Frenet formulas, curvatures, invariance theorems, curves with prescribed curvatures.


Unit 3. Plane Curves

Explicit formulas for plane curves, rotation number of a closed curve, osculating circle, evolute, involute, parallel curves, "Umlaufsatz". Convex curves and their characterization, the Four Vertex Theorem.


Unit 4. 3D Curves - Curves on Hypersurfaces

Explicit formulas, projections of a space curve onto the coordinate planes of the Frenet basis, the shape of curve around one of its points, hypersurfaces, regular hypersurface, tangent space and unit normal of a hypersurface, curves on hypersurfaces, normal sections, normal curvatures, Meusnier's theorem.


Unit 5. Hypersurfaces

Vector fields along hypersurfaces, tangential vector fields, derivations of vector fields with respect to a tangent direction, the Weingarten map, bilinear forms, the first and second fundamental forms of a hypersurface, principal directions and principal curvatures, mean curvature and the Gaussian curvature, Euler's formula.


Unit 6. Surfaces in the 3-dimensional space

Umbilical, spherical and planar points, surfaces consisting of umbilics, surfaces of revolution, Beltrami's pseudosphere, lines of curvature, parameterizations for which coordinate lines are lines of curvature, Dupin's theorem, confocal second order surfaces; ruled and developable surfaces: equivalent definitions, basic examples, relations to surfaces with K=0, structure theorem.


Unit 7. The fundamental equations of hypersurface theory

Gauss frame of a parameterized hypersurface, formulae for the partial derivatives of the Gauss frame vector fields, Christoffel symbols, Gauss and Codazzi-Mainardi equations, fundamental theorem of hypersurfaces, "Theorema Egregium", components of the curvature tensor, tensors in linear algebra, tensor fields over a hypersurface, curvature tensor.


Unit 8. Topological and Differentiable Manifolds

The configuration space of a mechanical system, examples; the definition of topological and differentiable manifolds, smooth maps and diffeomorphisms; Lie groups, embedded submanifolds in Rn, Whitney's theorem (without proof); classification of closed 2-manifolds (without proof).


Unit 9. The Tangent Bundle

The tangent space of a submanifold of Rn, identification of tangent vectors with derivations at a point, the abstract definition of tangent vectors, the tangent bundle; the derivative of a smooth map.


Unit 10. The Lie Algebra of Vector Fields

Vector fields and ordinary differential equations; basic results of the theory of ordinary differential equations (without proof); the Lie algebra of vector fields and the geometric meaning of Lie bracket, commuting vector fields, Lie algebra of a Lie group.


Unit 11. Differentiation of Vector Fields

Affine connection at a point, global affine connection, Christoffel symbols, covariant derivation of vector fields along a curve, parallel vector fields and parallel translation, symmetric connections, Riemannian manifolds, compatibility with a Riemannian metric, the fundamental theorem of Riemannian geometry, Levi-Civita connection.


Unit 12. Curvature

Curvature operator, curvature tensor, Bianchi identities, Riemann-Christoffel tensor, symmetry properties of the Riemann-Christoffel tensor, sectional curvature, Schur's Theorem, space forms, Ricci tensor, Ricci curvature, scalar curvature, curvature tensor of a hypersurface.


Unit 13. Geodesics

Definition of geodesics, normal coordinates, variation of a curve, the first variation formula for the length, . Gauss Lemma, description of geodesic spheres about a point with the help of normal coordinates, minimal property of geodesics.