微分幾何 III – 直線和 Geodesics
What is a straight line?
Local viewpoint –
1. no curvature – not working in curved space
2. constant curvature (S2 sphere) – same as curvature? but no torsion ?
3. any smooth curvature ?
1. minimum distance – integration of ds (function of metric), then apply Euler Lagrangian
2. local and global may not equal, depending on topologic shape – Global Gauss Bonnet ?
straight line – curvature is the same as ambient
constant curvature – S2
geodesic – curvature is the same as ambient?
In general with gij metric tensor mapping, and curvature k mapping
How to find the geodesic?
From the other (Global) viewpoint – minimum distance
metric integration – Euler Lagrangian method
should have the same result