d2xkdt2+Γijkdxidtdxjdt=0d squared x to the k-th power over d t squared end-fraction plus cap gamma sub i j end-sub to the k-th power d x to the i-th power over d t end-fraction d x to the j-th power over d t end-fraction equals 0
To illustrate this, consider a simple case: a 2D sphere where we want to find the shortest path between two points. In Riemannian geometry, these are "Great Circles." Why this is helpful: Riemannian Geometry.pdf
: Calculation of the symbols of the second kind, Γijkcap gamma sub i j end-sub to the k-th power d2xkdt2+Γijkdxidtdxjdt=0d squared x to the k-th power over
: A visual representation of the resulting manifold and the geodesics (shortest paths) between two user-defined points. Educational Visualization: Geodesic on a Sphere Riemannian Geometry.pdf
Introduction to Riemannian Geometry and Geometric Statistics - HAL-Inria
Since the "Riemannian Geometry.pdf" document likely covers the study of differentiable manifolds equipped with an inner product at each point, a highly useful feature for a student or researcher is a .