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:
Introduction to Riemannian Geometry and Geometric Statistics - HAL-Inria Riemannian Geometry.pdf
Riemannian geometry is famous for its complexity, often requiring students to manually compute Christoffel symbols and solve differential equations to find the shortest paths (geodesics) on a curved surface. This feature would automate those grueling steps. Useful Feature: Metric Tensor & Geodesic Visualizer This feature would allow you to input a metric tensor gijg sub i j end-sub and automatically generate the following: To illustrate this, consider a simple case: a
: Solving the second-order differential equation that describes the path of a particle in free fall: Useful Feature: Metric Tensor & Geodesic Visualizer This
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
: You can use it to check manual calculations for textbooks like M. Spivak's Calculus on Manifolds .