Research in differential geometry I am an 3rd year undergrad interested in mathematics and theoretical physics. I have been reading some classical differential geometry books and I want to pursue this subject further. I have three questions:
1) What are the current research topics in differential geometry? How is scope in those areas?
2) How should I go about  to pursue research in this area? I mean to say what background I should have if I want to start research in this area? How I should go about it i.e. what books  I should read and what are some important concepts?
3) As now I am reading classical differential geometry(Do' Carmo's book), how much time it will take me to start my research?
 A: Here is a partial answer based on my comment: 
At the very least, you should read the other do Carmo's book: "Riemannian Geometry". I do not know though if you have enough background to handle it. The book you are reading is basically about mathematics of 18th and 19th century. 
(It is good that you are reading it, keep doing so, but it prepares you for the current research about as much as reading about classical mechanics and electromagnetism prepares you for research in, say, string theory.) 
In contrast, the "R.G." is about math of 20th century, until mid 1960s. Without knowing at least the first 4 chapters of it, you will be facing basic linguistical problems understanding what the modern Differential Geometry is about. I can write about current research in geometric flows or in minimal surfaces, or in Kähler-Einstein metrics, but it would be mostly useless at this point. 
To do serious research in modern differential geometry you also need strong background in:


*

*Algebraic topology (say, to the extent covered in Hatcher's "Algebraic Topology" book). 

*Functional Analysis, Sobolev spaces, etc.

*Linear and Nonlinear PDEs (primarily elliptic and parabolic), at least if you will be doing geometric analysis. See for instance D. Gilbarg, N. Trudinger, "Elliptic Partial Differential Equations of Second Order".

*Possibly other fields, depending on the differential geometry subarea: Complex analysis, algebraic geometry, geometric topology, measure theory... 
How long would it take you to get the right background (determined by your future PhD advisor) to start research, is impossible to tell, it depends on too many factors.    
