Good problem book in riemannian geometry with specific exersices in curvature

Question

Please I'm looking for some exersices similar to this one below. I have looked in a lot of books but in vain. If some one can suggest some books or links, I would be very grateful.

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Let $(M,\langle,\rangle)$ be a riemannian manifold equipped with th connexion of Levi-Civita $\nabla$.

1. Show that in an orthonormal frame, $\Delta(f)=\sum_{i=1}^n Hess(f)(E_i,E_i)$
2. Show that $Ric(X,Y)=\sum_{i=1}^n \langle R(X,E_i)Y,E_i\rangle$

Let $X$ be a Killing vector field on $M$, i.e., $$X\langle Y,Z\rangle -\langle [X,Y],Z\rangle -\langle Y,[X,Z]\rangle =0, $$ and let $f :M\to \mathbb R$ be a function defined by : $f(p)=\frac{1}{2} \langle X(p),X(p) \rangle $

1. Show that $\langle \nabla_YX,Z\rangle +\langle Y,\nabla_Z X\rangle =0.$

2. Show that for any function $g$, $\Delta(X(g))= X(\Delta(g)).$

3. Show that $\nabla f=-\nabla_XX.$

4. Show that $Hess(f)(Y,Y)=-\langle R(X,Y)X,Y\rangle +\langle \nabla _YX,\nabla _YX\rangle .$

5. Show that $\Delta(f)=-Ric(X,X)+|\nabla X|^2.$

Answer 1

The following are more than what you want (The first one is contains all that you want):

• Petersen, Peter, Riemannian geometry, Graduate Texts in Mathematics 171. New York, NY: Springer (ISBN 978-3-319-26652-7/hbk; 978-3-319-26654-1/ebook). xvi, 401 p. (2016). ZBL1220.53002.

• Gadea, Pedro M.; Muñoz Masqué, Jaime; Mykytyuk, Ihor V., Analysis and algebra on differentiable manifolds: a workbook for students and teachers, Problem Books in Mathematics. London: Springer (ISBN 978-94-007-5951-0/hbk; 978-94-007-5952-7/ebook). xxv, 617 p. (2013). ZBL1259.53002.

• Lee, John M., Introduction to Riemannian manifolds, Graduate Texts in Mathematics 176. Cham: Springer (ISBN 978-3-319-91754-2/hbk; 978-3-319-91755-9/ebook). xiii, 437 p. (2018). ZBL1409.53001.