स्नातकोत्तर प्रथम वर्षGeometry & Topologyपाठ उपलब्ध
Riemannian geometry
Curved spaces studied from the inside: Riemannian metrics, the Levi-Civita connection, geodesics and the exponential map, curvature tensors, Jacobi fields, the global Gauss–Bonnet theorem, Hopf–Rinow, and comparison theorems such as Bonnet–Myers.
math.geom.riemannian-geometry
इस पर आधारित
43 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Define Riemannian metrics, the Levi-Civita connection and geodesics, and compute them in examples (spheres, hyperbolic space, surfaces of revolution).
- Define the curvature tensor, sectional and Ricci curvature, and interpret curvature through Jacobi fields and the spreading of geodesics.
- State and prove the global Gauss–Bonnet theorem for surfaces and apply comparison theorems such as Bonnet–Myers.
परिभाषा
- A Riemannian metric on a smooth manifold is a smoothly varying inner product on each tangent space.
- The Levi-Civita connection is the unique torsion-free connection compatible with the metric.
- A geodesic is a curve γ with ∇_{γ′} γ′ = 0; locally it minimises length.
- Sectional curvature K(σ) of a tangent 2-plane σ is the Gaussian curvature at the point of the surface swept out by geodesics tangent to σ.
प्रमेय
- Fundamental theorem of Riemannian geometry: the Levi-Civita connection exists and is unique.
- Gauss–Bonnet: for a compact oriented surface M without boundary, ∫_M K dA = 2π χ(M).
- Hopf–Rinow: for a connected Riemannian manifold, metric completeness is equivalent to geodesic completeness, and then any two points are joined by a minimising geodesic.
- Bonnet–Myers: a complete n-manifold with Ricci curvature ≥ (n − 1)k > 0 is compact with diameter at most π/√k and finite fundamental group.
आम भ्रांतियाँ
- Thinking curvature needs an ambient space; Gauss's Theorema Egregium shows the Gaussian curvature of a surface is intrinsic.
- Believing a geodesic is always the shortest path between its endpoints; on a sphere a great-circle arc longer than half the circle is a geodesic but not minimising.
आगे ले जाता है
वैकल्पिक खोजें
- Geometric analysis Ricci flow deforms a Riemannian metric by its Ricci curvature; Perelman used it to prove the Poincaré and geometrisation conjectures.
- Geometric group theory The fundamental group of a closed manifold of negative curvature is a hyperbolic group, and its large-scale geometry mirrors that of the universal cover (Švarc–Milnor).
- ← Surfaces, polyhedra and the Euler characteristic
- ← Non-Euclidean geometry: spherical and hyperbolic
अधिगम पथ
आगे पढ़ने के लिए
- Manfredo P. do Carmo, Riemannian Geometry (1992)
- John M. Lee, Introduction to Riemannian Manifolds (2018)