What is known about analogous results of Kazdan and Warner in higher dimensions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:05:59Z http://mathoverflow.net/feeds/question/118130 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118130/what-is-known-about-analogous-results-of-kazdan-and-warner-in-higher-dimensions What is known about analogous results of Kazdan and Warner in higher dimensions? Ritwik 2013-01-05T15:17:08Z 2013-01-05T15:17:08Z <p>First let me state a Theorem due to Kazdan and Warner: </p> <p>Let M be a compact two dimensional orientable manifold. Let $f: M \rightarrow \mathbb{R}$ be a function that has the same sign as $\chi(M)$, the Euler characteristic of $M$ at some point. Then $M$ admits a metric $g$ such that the Gaussian curvature $K$ is the given function $f$. ''</p> <p>(I am actually not sure if orientable is necessary). Is anything known about the following question: </p> <p>Let M be a compact, orientable manifold. Let $f: M \rightarrow \mathbb{R}$ be a function that has the same sign as $\chi(M)$, the Euler characteristic of $M$ at some point. Then does $M$ admit a metric $g$ such that $$*e(TM) = f$$ where $e(TM)$ is the Euler class of the Tangent bundle and * is the Hodge star operator?''</p> <p>Notice that in two dimensions this is precisely the Kazdan Warner theorem, since $$*e(TM) = \frac{K}{2 \pi}$$ </p>