# Exponential map

by brown042
Tags: exponential
 P: 11 Let q and q' be sufficiently close points on C^oo manifold M. Then is it true that any C^oo curve c:[a,b]-->M where c(a)=q, c(q)=q' can be represented as c(t)=exp$$_{q}$$(u(t)v(t)) where u:[a,b]-->R,v:[a,b]-->TM$$_{q}$$ and ||v||=1? My question comes from Chapter 9 corollary 16 and 17 of Spivak vol1. In the proof of corollary 17 I think he assumes this fact. Thanks.
 P: 11 Anyone has idea?
 Sci Advisor P: 2,341 First, if v is a vector field, which we consider as a first order partial linear differential operator on the ring of smooth functions on our Riemannian manifold, its exponential gives the family of integral curves, in the language of first order linear systems of ODEs. Anyone who has seen very many of my PF posts know that I am constantly yakking about integral curves. I used to also frequently mention the word "exponential", but more recently I've been trying to "dumb down" my posts. Why so many technical terms? Because there are different motivations for the various usages, and understanding how the notion of a vector field, in the modern theory of manifolds, unifies numerous apparently distinct concepts with venerable histories is crucially important! So tossing around all these terms can actually help those students who aren't frightened off. We are investigating Gaussian charts (introduced in his Oct 1827 paper) on some neighborhood of a point q. In Spivak's account, Lemma 15 constructs concentric "spheres" around q. Corollaries 16, 17 concern "local properties" of geodesic curves. What part of the proof wasn't clear?
P: 11

## Exponential map

So it was just smoothness of exponential map and ODE fact!.
Proof is clear now. So you don't like the word "exponential map" because it is a technical term?
I am interested in differential geometry and reading Spivak's book. Sometimes I wonder if it is necessary to study all the materials in the book.
If you have any more personal opinion on studying differential geometry or topology, I would like to hear it!
Thanks.
P: 2,341
 Quote by brown042 So you don't like the word "exponential map" because it is a technical term?