Misha Verbitsky
INTRODUCTION TO DIFFERENTIAL GEOMETRY AND GEOMETRIC ANALYSIS
Spring 2013, Math in Moscow and HSE.
Differential geometry is the study of smooth
manifolds by means of vector bundles and the
Lie group theory. I will give a gentle introduction
to some of the most basic notions of differential
geometry: vector bundles, tangent spaces, sheaves,
connections and differential operators.
Approximate syllabus.
1. Smooth manifolds, partition of unit,
Hausdorff dimension and Hausdorff measure.
Whitney embedding theorems.
2. Sheaves, categories, limits, colimits, and
germs of functions. Smooth manifolds as ringed
spaces.
3. Derivations on the ring of smooth functions;
vector fields as derivations. Vector bundles;
equivalence of different definitions of vector
bundles. Serre-Swan theorem.
4. Differential operators and their symbols.
De Rham algebra and de Rham differential.
Lie derivative and Cartan's formula.
5. Elliptic equations and their properties.
Weak maximum principle. Harmonic functions,
mean value property of harmonic functions.
6. Stokes' theorem, de Rham cohomology,
applications to topology.
7. Definition of a connection.
Construction of connections on vector bundles.
Parallel transport along a connection.
8. Torsion and curvature of a connection.
Existence and uniqueness of the Levi-Civita
connection.
Prerequisites:
some knowledge of linear algebra (tensor product,
polylinear, symmetric, anti-symmetric forms,
self-adjoint and anti-self adjoint operators)
analysis (manifolds, coordinates, Taylor series,
partition of unit) and topology (topological spaces,
continuous maps, limits, compactness).