Budapest Semesters in Mathematics

Lecture
Notes by **Balázs Csikós**

**CONTENTS**

**Unit
1. Basic Structures on R ^{n}, 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 R ^{n}; balls, open subsets,
the standard topology on R^{n}, 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 R ^{n},
the osculating flag, vector fields, moving frames and Frenet frames along
a curve, orientation of a vector space, the standard orientation of R^{n},
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 R ^{n}, Whitney's theorem (without
proof); classification of closed 2-manifolds (without proof).**

**Unit
9. The Tangent Bundle**

**The
tangent space of a submanifold of R ^{n}, 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.**