I spent months working on a meaningless problem. And what's worse is I fail to prove it. Although I am rookie, I want to know the prevailing direction of Riemannian geometry. Besides, the Poincare conjecture has been proved for more than decade. And seemly, many problems that can be solved by Ricci flow have also been solved. And seemly, Ricci flow is weak for 4-dim problems. So, is it worth spending a lot of time on Ricci flow now ?
I don't know how a trivial math practitioner should spend his life. And I don't like teaching. So I felt lost. I was hoping to get some advice. Thanks.