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.
Calculus on Manifolds by Spivak is a quality introduction if you haven't seen a lot of Analysis.