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I just jumped into a project related to an estimation algorithm. It needs to build measures between two distributions. I found a lot of papers in this field required a general idea from differential geometry, which is like a whole new area for me as a linear algebra guy. I indeed follow a wiki leading studying by looking up the terms, and start to understand some of them, but I found this way of studying is not really good for me, because it is hard to connect these concepts.

For example, I know the meaning for concepts like Manifold, Tangent space, Exponential map, etc. But I lack the understanding why they are defined in this way and how they are connected.

I indeed want to put as much effort as needed on it, but my project has a quick due time, so I guess I would like to have a set of the minimum concepts I need to learn in order to have some feeling for this field.

So in short, I really want to know if there is any good reference for beginner level like me -- for engineering background student? Also since my background is mainly in linear algebra and statistics, do I have to go through all the materials in geometry and topology ?

I really appreciate your help.

Following up: I checked the books suggested by answerers below, they are all very helpful. Especially I found Introduction to Topological Manifolds suggested by kjetil is very good for myself. Also I found http://www.youtube.com/user/ThoughtSpaceZero is a good complement (easier) resource that can be helpful for checking the basic meanings.

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    $\begingroup$ I think this question is too broad to get a good answer here. Did you read the relevant wikipedia articles, and/or look at a suitable textbook? $\endgroup$ – user64687 Jul 10 '13 at 21:46
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    $\begingroup$ As stated, this is more or less impossible to answer in any way but «go read a bunch of relevant textbooks», really. $\endgroup$ – Mariano Suárez-Álvarez Jul 10 '13 at 21:47
  • $\begingroup$ @AsalBeagDubh and Mariano : thank you for your help. I am sorry if the question is too broad, that's because I don't even know how big the picture is for this field, so the question jumped between things far apart. I modified the question to ask for reference for students with background in linear algebra and statistics $\endgroup$ – Jack2019 Jul 11 '13 at 14:45
  • $\begingroup$ (BTW, a manifold is not really a "collection of distributions"; you will have to forget some of the wrong things before you start learning the right ones.) $\endgroup$ – Willie Wong Jul 11 '13 at 14:57
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I suggest you read Lee's introduction to topological manifolds followed by his introduction to smooth manifolds.

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  • $\begingroup$ Thanks a lot. I found I can follow Introduction to Topological Manifold without too much problem. I've done Chapter 1 and Chapter 2 so far. $\endgroup$ – Jack2019 Jul 12 '13 at 16:45
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For a very quick introduction to thinking about manifolds, I recommend for you Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Try reading through the whole thing (it is less than 200 pages!) with focus on chapter 5.

After that, perhaps a good place to continue would be either Do Carmo's Riemannian Geometry or John M Lee's Riemannian Manifolds: an introduction to curvature.

Then, you can probably pick up individual topics pretty clearly from the Wikipedia articles or the references that they cite.


A quicker way, if you work at an academic institution, is to pay a mathematics graduate student to explain what you need (show him or her the literature you are trying to read so he or she knows what to cover) to you. This way you won't get a comprehensive picture, but will likely know just a little bit more than enough to do your project.

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There is a relatively new book about differential geometry and some related concepts, written for engineering types: Gregory S. Chirikjian, Stochastic Models, Information Theory, and Lie Groups, Volume 1 Classical Results and Geometric Methods (The second volume have also appeared)

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