Abstract
We discover following analytic / geometric properties on Riemannian foliations with bundle-like metric and totally geodesic leaves, or shortly, totally geodesic foliations. Under a certain curvature condition, we obtain (1) Sobolev-isoperimetric inequalities, global Poincar\'e inqualities, and a lower bound for Cheeger's isoperimetric constant, (2) Poincar\'e inequalities on balls and uniqueness of positive(or $L^p,p\geq 1$) solutions for the subelliptic heat equation, (3) A lower bound for the first non-zero eigenvalue of sub-Laplacians (Lichnerowicz theorem), and Obata's sphere theorem. In this context, the curvature condition is a sub-Riemannian analogue of lower bounds for Ricci curvature tensor. Earlier, it is given by Baudoin-Garofalo's curvature dimension inequality, or Baudoin's Weitzenb"ock formulas for one forms. Our framework includes CR Sasakian manifolds with Tanaka-Webster (or pseudo-Hermitian) Ricci tensor bounds, K-contact manifolds, and Carnot group of step 2.
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
Committee Chair
Fabrice Baudoin
Date of Award
Spring 2015
Recommended Citation
Kim, Bumsik, "Functional inequalities and the curvature dimension inequality on totally geodesic foliations" (2015). Open Access Dissertations. 487.
https://docs.lib.purdue.edu/open_access_dissertations/487
First Advisor
Fabrice Baudoin
Committee Member 1
Rodrigo Bañuelos
Committee Member 2
Laszlo Lempert
Committee Member 3
Sai Kee Yeung