Sentences

1. which is ..., since ... and, thus, ... .

2. The zero set of a Killing field is a disjoint union of totally geodesic submanifolds each of even codimension.

3. We can look at transversally harmonic maps in another light.

4. Based on the previous estimates various monotonicity formulas are given and then specialized to the case of arbitrary compact Riemannian and Einstein manifolds.

5. the results are compared with different gradient estimates by Yau,its improved version by Bakry and Qian, and with an estimate by Hamilton.

6. By going through Li–Yau’s paper carefully, one can get similar results on the estimates of Greene’s function, ...

7. ... is called an infinitesimal automorphism (i.a. for short) of...

8. The proof of Theorem 1.7.1, with S replaced by −S,now goes through essentially word for word to show that both S and R must vanish identically.

9. Because all our considerations here are of a purely local nature we can ...

10. To this aim, we shall consider....

11. Of special interest is of course the case, where ...

12. Working in a more general context, we look for ...