Question: I am looking to learn about manifolds for use in signal processing. I have a engineering degree where I have covered calculus and basic linear algebra, with this background in mind, does anyone have a good recommendations for a introductory textbook on manifolds?

Answer: 
If you want an elementary introduction to manifolds, assuming very little background, I'd suggest you start with 

* [***Differential Geometry: Curves - Surfaces - Manifolds, Second Edition***](http://rads.stackoverflow.com/amzn/click/0821839888) by Wolfgang Kühnel.

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***Then (or else)***  

You might want to look into one or more of the following: 

* [***Introduction to Smooth Manifolds***](http://rads.stackoverflow.com/amzn/click/1441999817) by John M. Lee. (It's a very well-known, loved, and highly recommended text.)

You might want to compare Lee's book (table of contents, etc) with 

* J. Munkres's text [***Analysis on Manifolds***](http://rads.stackoverflow.com/amzn/click/0201315963) and/or
* Spivak's [***Calculus on Manifolds***](http://rads.stackoverflow.com/amzn/click/0805390219).

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You can preview each of the texts linked above at Amazon.com to see which best meets your needs, in terms of topics covered, apparent level of difficulty, etc.
Beware - Spivak is very pretty but very concise; skimming Munkres' Analysis on Manifolds suggests it covers much of the same material but in a much more user-friendly way.