which book,or paper I should read to learn and doing some work on isometry of surfaces?

Question

For research on Differential Geometry of curves and surfaces I read only Elementary Differential Geometry by A.N. pressley. I am trying to do something on Isometry of surfaces. Further which book,or paper I should read, please give me some suggestion?

Answer 1

In general isometries are a part of Riemannian Geometry, which is the study of manifolds with the extra structure of an inner product on each tangent space. If you really want to study research papers on isometries then you have to learn the language of Riemannian Geometry.

On the other hand, there is plenty to be done without going deep into differential geometry. Have you read about the Gauss-Bonnet Theorem? Or why there is not an isometry between the sphere and the plane? I am not sure about Pressley's book, but Do Carmo and Tapp do touch these topics. Furthermore Tapp develops some general notions of Riemannian Geometry (like geodesics) through concrete examples in $\mathbb{R^3}$

P.S. Although I suggested 2 books, I would advice to stick in the book you have started reading (in that case Pressley) and after finishing that go for another book.