I am teaching an introductory course in Riemannian geometry this semester. I am following do Carmo's book for the course. As a continuation of this course, I am planning to offer another course in Rimannian geometry for the next semester. My intention is to give proofs of theorems which are not there in do Carmo's book but can be understood with the knowledge of this book.
Therefore I am looking for theorems and proofs of Riemannian geometry which are not there in introductory textbooks but requires only introductory knowledge of Riemannian geometry to understand them.
Any kind of suggestions or references of books, monographs, papers would be extremely helpful. Thank you in advance.