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The Geometry of Physics by Theodore Frankel

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CHAPTER 10

Geodesics

image

How rapidly do nearby geodesics separate?

10.1. Geodesics and Jacobi Fields

10.1a. Vector Fields Along a Surface in Mn

Let x:Uimage2Mn be a differentiable map of a rectangle in the plane into Mn. We call this map a (parameterized) surface even though we put no demands on the rank of the differential x*; that is, x/∂u1 and x/∂u2 may be dependent.

image

Figure 10.1

Let us again put u1 = u and u2 = ν.

A smooth map v: U2M that ...

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