The student knows how to use the following notions, both in theoretical proofs, as well as in explicit computations:
- smooth vector bundle
- Riemannian manifolds
- connections on vector-bundles and the Levi-Civita connection
- geodesics
- parallel transport
- the various notions of curvature on a Riemannian manifold
- Jacobi fields
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Riemannian geometry was born in Bernhard Riemann's famous habilitation lecture on the foundations (hypotheses) of geometry in 1854. By defining a Riemannian metric on a manifold as a smoothly varying inner product on the tangent space at any point, we can introduce local concepts such as length, angle, area, volume and curvature, and thus we can extend concepts from classical Gaussian differential geometry (i.e., surfaces in a three-dimensional Euclidean space) to abstract (i.e., no longer embedded in a certain Euclidean space) manifolds of arbitrary dimension. Global geometric quantities can be obtained using integration.
This introductory course covers all basic notions of Riemannian geometry. It is strongly recommended to students interested in geometry and/or mathematical physics. For instance, the indefinite variation of Lorentzian geometry is the language needed to understand Einstein's general theory of relativity about the structure of our universe.
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The course Manifolds in the previous semester is an absolute must, without which this course will be incomprehensible. Useful but not absolute necessary is the second year course on Curves and Surfaces in a three dimensional Euclidean space. |
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Oral exam and assignments
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