Modern Geometry · Mathematics
Jacobi fields, second fundamental form, embeddings of manifolds, spaces of constant sectional curvature, variations of energy, mathematical GR -- Schwarzschild, Kerr black holes, trapped surfaces.
Knowledge: A survey of basic differential geometry (smooth manifolds, differential forms, Riemannian manifolds)
Jacobi Fields, Second fundamental forms, Hadamard, Variations, and most fun, General Relativity.
Differential Topology(manifolds, tangents, derivations, vector and tangent bundles, tensors, vector fields), Riemannian Geometry(metrics, curvature, lie groups, geodesics)
I learned about manifolds, bundles, complex curves, de Rahm cohomology, and Riemannian geometry among other special topics in modern geometry.
In this class we studied Jacobi fields, isometric immersions, completeness of Riemannian manifolds, spaces of constant curvature, and variations of energy of curves. We also got a crash course introduction to mathematical General Relativity, regarding the initial value formulation of the Einstein field equations, the causal structure of spacetime, null-structure equations, black holes, and wave-equations/energy estimates on black hole spacetimes.
Need to have proper background to take the class
Applying the concepts of connections and curvature of Riemannian manifolds studied in the first semester of this course. Jacobi Fields, Isometric Immersions, Geodesically Complete Manifolds, Spaces of Constant Curvature, Variations of Energy. The class also ended with an overview of Mathematical GR.
Smooth manifolds, fiber bundles (chiefly vector bundles like the tangent bundle and the tensor bundles), local operations like Lie derivative and exterior derivative, connections on vector bundles, Riemannian geometry like the Levi-Civita connection and the different curvature tensors, Riemannian geometry and differential geometry of Lie groups, some focus on quotient manifolds and their geometry.
Survey course in Modern Geometry - as advertised. Covered manifolds, differential geometry, complex geometry, with memorable forays into algebraic topology, category theory, and general relativity.