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Пишет Misha Verbitsky (![[info]](http://lj.rossia.org/img/userinfo.gif){kind=small} tiphareth)@ 2023-04-11 10:30:00 |
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| Настроение: |  sick |
| Музыка: | Ryuichi Sakamoto - Left Handed Dream |
| Entry tags: | math |
программа экзамена по дифференциальной геометрии
Написал программу аспирантского экзамена по
высокоуровневой дифф. геометрии. Опустил практически
все, что требует эллиптических уравнений, теоремы об
индексе, групп Ли и хар. классов, это еще примерно
столько же; также опустил почти все, что касается
оснований и анализа на многообразиях.
Differential geometry
1. Connections in vector bundles. Ehresmann connections.
Principal bundles and associated vector bundles. G-structures
on manifolds. Spin-structure and its existence.
2. Lie derivative, Cartan formula, de Rham differential
expressed in terms of commutators and Lie derivatives.
Torsion of a connection. Intrinsic torsion of a G-structure.
3. Riemannian structures. Levi-Civita connection,
its existence and uniqueness. Symmetries of the
curvature tensor. Decomposition of the curvature
tensor onto Ricci curvature, scalar curvature
and Weyl curvature. Decomposition of the curvature
tensor in dimension 4. Self-dual and anti-self-dual
4-manifolds and their twistor spaces.
4. Geodesics, completely geodesic submanifolds,
Hopf-Rinow theorem. Properties and applications of
the exponential map. Sectional curvature and the
curvature pinching. Hadamard-Cartan theorem and
Myers theorem. Gromov's almost flat manifolds.
5. Geometric properties of the Ricci curvature.
Bishop-Gromov inequality and Gromov's compactness
theorem.
Literature:
S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry
Arthur L. Besse, Einstein Manifolds
Simon Salamon, Riemannian geometry and holonomy groups
Manfredo do Carmo, Riemannian Geometry
Peter Petersen, Riemannian geometry
Loring Tu, Differential Geometry: Connections,
Curvature, and Characteristic Classes
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