M-Curves: Representations & Properties of C^oo Manifolds

[brown042]

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.

 

[brown042]

Anyone has idea?

 

[Chris Hillman]

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?

 

[brown042]

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.

 

[Chris Hillman]

So you don't like the word "exponential map" because it is a technical term?

I didn't say any such thing! Please reread what I wrote

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 are goal is to master differential geometry, sure that (five-volume) book and many more.

If you have any more personal opinion on studying differential geometry or topology, I would like to hear it!

It's fun?

 

[quasar987]

If I may, I would like to use this thread to ask you a question Chris. I am aware that manifolds pop up in various branches of math a physics, but when does DIFFERENTIAL GEOMETRY on those manifold occur?

General relativity aside, what is a practical application of the lie derivative? Of the covariant derivative? Of Paul-Levi's Isoperimetric inequality. You get the idea.

I'm sure there are many, but I would like to get specific examples. Thanks!

 

[Chris Hillman]

Much of the theory of topological manifolds would fall outside the scope of differential geometry, but possibly within the scope of differential topology. The theory of smooth manifolds would probably be considered to belong to the huge field of differential geometry. The theory of manifolds includes both topological and smooth manifolds. These are all somewhat informal terms, and for many other reasons it's impossible to precisely pigeonhole any of these topics!

Practical applications of Lie derivative are too numerous to list, but for example Lie's theory of symmetry of differential equations has highly applicable applications in applied mathematics as well as pure mathematics (e.g. Cartan's far-reaching program for classifying virtually kind of "geometric" structures which can be placed on a smooth manifold). Similarly for covariant derivative, but to mention something which comes up quite a bit, this has applications to the theory of robot motion planning and to robotic vision. And without inequalities, mathematics would be stuck in a very primitive stage indeed.