I just jumped into a project related to an estimation algorithm. It needs to build measures between two distributions. I found a lot of papers in this field required a general idea from differential geometry, which is like a whole new area for me as a linear algebra guy. I indeed follow a wiki leading studying by looking up the terms, and start to understand some of them, but I found this way of studying is not really good for me, because it is hard to connect these concepts.
For example, I know the meaning for concepts like Manifold, Tangent space, Exponential map, etc. But I lack the understanding why they are defined in this way and how they are connected.
I indeed want to put as much effort as needed on it, but my project has a quick due time, so I guess I would like to have a set of the minimum concepts I need to learn in order to have some feeling for this field.
So in short, I really want to know if there is any good reference for beginner level like me -- for engineering background student? Also since my background is mainly in linear algebra and statistics, do I have to go through all the materials in geometry and topology ?
I really appreciate your help.
Following up: I checked the books suggested by answerers below, they are all very helpful. Especially I found Introduction to Topological Manifolds suggested by kjetil is very good for myself. Also I found http://www.youtube.com/user/ThoughtSpaceZero is a good complement (easier) resource that can be helpful for checking the basic meanings.