Lecture 11 (Jan 16, 2017)

Sergei Yakovenko's blog: on Math and Teaching

Geodesics

Definitions of geodesic curves on a Riemmanian manifold. Differential equations of the second order. Local existence of solutions. Geodesic map. Geodesic spheres, orthogonality. Local minimality of geodesic curves. Metric and geodesic completeness of Riemannian manifolds.

Survey of adjacent areas. Behavior of the nearby geodesics. Jacobi field. On surfaces: conjugated points, the role of the Gauss curvature, global properties of manifolds with positive and negative curvature (comparison). Hyperbolic plane $latex mathbb H$, geodesics on it. Realization of the hyperbolic geometry on a surface in $latex mathbb R^3_{++-}$. Impossibility of global embedding of $latex mathbb H$ into $latex mathbb R^3_{+++}$.

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