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April 11th, 2023
| April 11th, 2023 | |
|---|---|
| 10:30 am [Link] |
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 Привет Current Mood: |
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