The title of the book

" A first course in general relativity" by B. Schutz

was interesting for me.

I don't have a book for you, but start with two adjacent articles in one journal:

Tenenbaum, Joshua B., Vin De Silva, and John C. Langford. "A global geometric framework for nonlinear dimensionality reduction." Science 290.5500 (2000): 2319-2323.

http://www.robots.ox.ac.uk/~az/lectures/ml/tenenbaum-isomap-Science2000.pdf

Roweis, Sam T., and Lawrence K. Saul. "Nonlinear dimensionality reduction by locally linear embedding." Science 290.5500 (2000): 2323-2326.

http://locally-linear-embedding.googlecode.com/svn-history/r27/trunk/papers/nonlinearDimensionalityReductionByLocallyLinearEmbedding.pdf

Enjoy!

The title of the book

" A first course in general relativity" by B. Schutz

was interesting for me.

"Differential Topology" by Guillemin and Polark.

http://books.google.co.in/books?id=FdRhAQAAQBAJ&printsec=frontcover&dq=differential+topology+ams&hl=en&sa=X&ei=KZVcU47nO4WOrQfH_YG4Dw&ved=0CDcQ6AEwAg#v=onepage&q=differential%20topology%20ams&f=false

"Elementary Differential Geometry" by By Barrett O'Neill

http://books.google.co.in/books?id=OtbNXAIve_AC&printsec=frontcover&dq=differential+geometry+neil&hl=en&sa=X&ei=yJVcU_eWE46RrAfC74DQBA&ved=0CCwQ6AEwAA#v=onepage&q&f=false

Hope, it helps you. Thanks.

I will suggest,

M. do Carmo, "differential geometry of curves and surfaces"

because it is a classic in differential geometry and good if you are a beginner in the subject. In fact he does not use the tensor notation and does everything with sums.

It depends on your level and time, I think that do Carmo did a very nice work, I also like the Spivak vol II. The first three chapters will give you a lot ideas. The book of Guillemin and Polark is related to the very related field of differential topology.

Dear Shakeel

I can give you here couple of suggestions. I also noticed that some people adviced books which deal more on 'analysis on manifolds' and 'differential topology'. If you have not taken at least 2 good courses on topology and advanced calculus + perhaps preferably course on algebraic topology, I strongly suggest that you don't start your diff.geom studies with such books !

Here are couple of books whose background expectations should not be so overwhelming !

Believe me I kept an 8 credit course on diff.geometry in spring 2012 and I think I

started with too harsh material. So I have made that mistake,don't repeat that :-).

1) Differential Geometry of Curves and Surfaces, by Manfred do Carmo.

2) Riemannian Geometry: A Beginners Guide, Second Edition, by Frank Morgan

3) This is little bit more challencing: Differential Geometry: Curves - Surfaces - Manifolds, by Wolfgang Kühnel.

4) Elementary Differential Geometry, (Springer Undergraduate Mathematics Series), by Pressley, A.N.

kind regards

Samuli Piipponen

University of Eastern Finland

I will suggest this books :W. Kuhnel, Dierential Geometry: Curves { Surfaces { Manifolds and M. P. do Carmo, Dierential Geometry of Curves and Surfaces, and also J. Oprea, Dierential Geometry and Its Applications