Riemannian Geometry

1-parameter 2-plane a₁ assume atlas c(to c₁ canonical closed curves closed geodesic compact conjugate points consider constant curvature convex coordinates covariant derivation critical point defined Definition denote diffeomorphism differentiable mapping E₁ eigenvalue equation exists fibre finite flow line follows geodesic c(t geodesic flow Hence Hilbert space homeomorphic homotopy hyperbolic isometry isomorphism Jacobi field Jacobi field Y(t K₁ Klingenberg Lemma length linear M₁ minimizing geodesic morphism open neighborhood orthogonal P₁ Proof Proposition radius Remark Riemannian manifold Riemannian metric scalar product sectional curvature sequence simple closed geodesics simply connected sphere submanifold subspace surface symmetric space symplectic t₁ T₁M tangent bundle tangent space tensor Theorem topology totally geodesic u₁ u₂ vector bundle vector field x₁ Y₁ Κο