A conformal integral invariant on Riemannian foliations

Details A conformal integral invariant on Riemannian foliations A conformal integral invariant on Riemannian foliations

2011-11-27

arxiv.org/abs/1111.6260v1

Mathematics

Differential Geometry

10 pages

Scientific paper

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 $\int_MY(R^T_{g^T})d\mu_g$ is independent of the choice of $g\in [g_0]_B$, where $g^T$ is the transverse metric induced by $g$ and $R^T$ is the transverse scalar curvature. Moreover if $q\geq 3$, we have $\int_MY(R^T_{g^T})d\mu_g=0$ 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 $\int_MY(R^T_{g^T})d\mu_g\neq 0$. Therefore, it is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension 2.

Affiliated with

Also associated with

No associations

Rating

A conformal integral invariant on Riemannian foliations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A conformal integral invariant on Riemannian foliations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A conformal integral invariant on Riemannian foliations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-687218