How to study to learn differential geometry for applying it to statistics Basically I want to learn information geometry or specifically the application of differential geometry in statistics to do a project. I am from a statistical background and have a knowledge about real analysis, several variable calculus, linear algebra.
One of my professors told me that the first three chapters from Do Carmo's Differential geometry would be sufficient.
Can someone assure me if that's enough or do I need to learn Riemannian geometry.
And If I need to learn Riemannian geometry then what should be my path for learning.
I don't want to learn rigorous mathematics. I just want to apply it to statistics.
 A: Avishek, not easy to answer with the little context you provided.
I would go first with what your prof said, and yes, Do Carmo is the place to go.
There, you will learn all about surfaces in $R^n$, which is basically classical differential geometry.
If, on the other hand, your project is at the research level (say master thesis or beyond), then download this article. That has to do with abstract information geometry, which in turn relies on modern differential geometry: manifolds, tensor calculus, etc. Basically, the chief difference between the first and the second is that in manifold theory you do not start from embedded manifold, rather you define the entire machinery intrinsically.
If you do not know classic geometry of surfaces, you still have to spend a few days on Do Carmo. Then prepare for a lot of sweat, to get into the modern approach.
Hope it helps
A: I think Do Carmo is a good option. Personally,  I'm a fan of John Lee's Introduction to Smooth Manifolds and its sequel Riemannian Manifolds. While these are written at a higher level, they really emphasize the geometric picture at work.
I think the survey by Nielsen is a good article and I found it very helpful to get a broad overview of IG. However, I would not recommend using it to learn differential geometry. Most books about information geometry take a very idiosyncratic approach to geometry, which can give rise to various misunderstandings. These are not a big deal if you are already familiar with differential geometry but are more of a problem if you are trying to learn it.
Both of these works are well worth reading if you are interested in IG, but I'll give an example of what I mean. Both Amari's book and the survey article by Nielsen state that the holonomy of a flat connection is trivial (although they don't use this language). In information geometry, the flat connections of interest are generally on exponential families (where this ends up being true). However, in general, the holonomy of a flat connection is not zero (it's induced by fundamental group). Furthermore, for this result, the connection must be both curvature- and torsion-free (not just curvature-free). Statistical manifolds are generally taken to have torsion-free connections, so this is not an issue in applications. These are relatively minor points if you are familiar with differential geometry, but would be misleading for someone learning it.
