|
- Which is the proper definition of a geodesic curve?
Geodesics are locally length minimizing, in that any sufficiently small portion is the shortest path between the endpoints of that portion
- differential geometry - geodesic in metric space and in manifolds . . .
In general "differential geometry geodesics" are only locally length minimizing; consider on the sphere the path from the north pole to the south pole, and then go a little further along this path This is a geodesic, but the quicker path would have been to go the other way around the sphere
- Geodesics on a cylinder: straight line or helix?
5 It is known that geodesics on a cylinder are helical lines (helix) As a special case of a helix--- straight line But one can take two points so cleverly that you can draw two helixes between them (see picture), one of which is a straight line Both satisfy geodesics differential equation
- The Geodesics of a Sphere. - Mathematics Stack Exchange
The Geodesics of a Sphere Ask Question Asked 11 years, 5 months ago Modified 3 years, 11 months ago
- Geodesics: From Affine Connection to Length Minimisation
For Q2: Not every affine connection preserves a Riemannian metric, so you cannot go from geodesics defined by (*) to Riemannian geodesics
- Geodesics on torus - Mathematics Stack Exchange
There is an excellent description of geodesics on a torus in this document
- differential geometry - why are geodesics invariant under isometries . . .
The most straightforward answer to your question at that point in DoCarmo is to note that geodesics are solutions of a system of differential equations (with Christoffel symbols appearing as the coefficients) Since the first fundamental form is — by definition — preserved by a (local) isometry, so will be the Christoffel symbols (because they are given in terms of the first fundamental
- differential geometry - Geodesics on paraboloid self-interesect in an . . .
Geodesics on paraboloid self-interesect in an infinite number of points Ask Question Asked 5 years, 7 months ago Modified 1 year, 11 months ago
|
|
|