Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A conformal integral invariant on Riemannian foliations (1111.6260v1)

Published 27 Nov 2011 in math.DG and math.AP

Abstract: Let $M$ be a closed manifold which admits a foliation structure $\mathcal{F}$ of codimension $q\geq 2$ and a bundle-like metric $g_0$. Let $[g_0]B$ be the space of bundle-like metrics which differ from $g_0$ only along the horizontal directions by a multiple of a positive basic function. Assume $Y$ is a transverse conformal vector field and the mean curvature of the leaves of $(M,\mathcal{F},g_0)$ vanishes. We show that the integral $\int_MY(RT{gT})d\mu_g$ is independent of the choice of $g\in [g_0]B$, where $gT$ is the transverse metric induced by $g$ and $RT$ is the transverse scalar curvature. Moreover if $q\geq 3$, we have $\int_MY(RT{gT})d\mu_g=0$ for any $g\in [g_0]B$. However there exist codimension 2 minimal Riemannian foliations $(M,\mathcal{F},g)$ and transverse conformal vector fields $Y$ such that $\int_MY(RT{gT})d\mu_g\neq 0$. Therefore, it is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension 2.

Summary

We haven't generated a summary for this paper yet.