A good start for: "Riemannian Manifolds" and "Spectral Riemannian Geometry" "This is the revision of the previous question"
As a grad student from another field of study (signal processing) rather than mathematics, I am eager to learn some introductory parts on "Riemannian Manifolds" and "Spectral Riemannian Geometry". Those are elegant topics in math and so needs some background to start. I'd like to know what are the pre-requisites for these topics in a detailed format (each pre-requisite with its nice source for that). There are some sources in the first answer, but I need to know in more details; I mean all the necessary backgrounds for above topics.
I appreciate if any one can introduce good starting path for this topic. 
 A: The book 

M. Berger, A Panoramic View of Riemannian Geometry

contains a readable summary of many of the big ideas in Riemannian geometry, for people who have not mastered the subject, and the only summary I know of to spectral geometry which is accessible with only a solid vector calculus background. Of course, as in any summary of advanced material, you will not have a clear and precise understanding of the material. To fill in the details, you will need to work through one of the many excellent Riemannian geometry textbooks (Do Carmo, Spivak, Petersen, Lee, or others). But Berger's book is one of a kind.
A: You should probably be familiar with at least some point-set topology. Allen Hatcher's notes should be sufficient. You can find them here: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf . Your analysis background should probably consist of some multivariable analysis – so that you understand the Implicit and Inverse function theorems in Euclidean space. As I recall, Pugh's book on Real Analysis covers these subjects nicely.
Now, if you are a great fan of multivariable calculus and multivariable analysis and want to build up geometric intuition, you should check out Guillemin and Pollack's Differential Topology. This book has definite limitations in that the treatment is rather outdated. However, you will get a nice intuition for manifold theory.
Outside of this, there are two main choices in my opinion. Lee's Introduction to Smooth Manifolds is comprehensive and easy to read, while being packed with good exercises. Tu's An Introduction to Manifolds is substantially less comprehensive, but equally easy to read. That it is shorter can be an advantage, depending on what your goal is. If you intend to cover the material in Differential Forms in Algebraic Topology by Bott and Tu, this book is the essential prerequisite. 
