Mathematics / advanced
Riemannian metrics and curvature
Connect metric tensors to connections, geodesics, curvature, comparison theorems, and global geometric structure.
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Module 01 advanced
Riemannian metrics and curvature
Connect metric tensors to connections, geodesics, curvature, comparison theorems, and global geometric structure.
- 01
Riemannian metrics and length functionals
Turn smoothly varying tangent-space inner products into invariant curve lengths, energy, and an intrinsic distance.
28 min - 02
Levi-Civita connection and parallel transport
Differentiate tangent fields without comparing unrelated tangent spaces directly, and track metric-preserving transport in a moving frame.
29 min - 03
Geodesic equations and exponential maps
Derive zero covariant acceleration, solve the local initial-value problem, and distinguish the exponential map's local invertibility from global uniqueness.
29 min - 04
Riemann curvature and sectional curvature
Fix a consistent curvature sign, compute a coordinate-independent sectional quantity, and distinguish a nonzero connection coefficient from actual curvature.
30 min - 05
Jacobi fields and geodesic variation
Differentiate a family of geodesics, retain covariant derivatives and the curvature sign, and locate conjugate points in constant curvature.
30 min - 06
Hopf–Rinow and geodesic completeness
Match metric and geodesic completeness under the correct manifold hypotheses, and separate existence of minimizing routes from completeness.
29 min - 07
Gauss–Bonnet on surfaces
Combine interior curvature, oriented boundary turning, and corner angles to recover the Euler characteristic of a compact surface.
30 min - 08
Ricci curvature and volume comparison
Trace sectional effects into Ricci curvature, state normalized Bishop–Gromov comparison, and compute a spherical volume ratio.
30 min - 09
Sectional curvature and comparison geometry
Use Jacobi-field growth to understand nonpositive curvature, then separate local curvature control from global topological uniqueness.
30 min - 10
Hodge Laplacian and harmonic forms
Use the metric to define the adjoint of exterior differentiation and identify unique harmonic representatives of cohomology on compact boundaryless manifolds.
31 min