微分幾何 III – 直線和 Geodesics

by allenlu2007

 

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 ?

 

Global viewpoint

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 

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