Riemannian manifolds with a Ricci lower bound is an important researched area of differential geometry, in particular in the context of comparison theorems. One result is a way to characterize Ricci curvature bounds through a condition of the Laplacian: the curvature-dimension inequality $\Gamma_2(f)\geq \frac{1}{n}(\Delta f)^2+\rho\Gamma(f)$ where $n$ is the dimension of the manifold, $\Gamma_2$ and $\Gamma$ two operators based on $\Delta$, the Laplacian and $\rho$ the bound of the Ricci curvature. Following this analytic, but equivalent, approach it is possible to find a generalization, which hing only on differential operator, also for non Riemannian cases where the Ricci curvature is not completely defined. In this thesis we study a generalization due to Baudoin and Garofalo for the case of sub-Riemannian manifolds with transverse symmetries. In Chapter 1 we set down the basic concepts we need throughout the thesis. Moreover in Chapter 2 we present the Riemannian curvature-dimension inequality and answer the question: what can be said about manifolds with a Ricci lower bound? Chapter 3 is entirely dedicated to introduce notions of sub-Riemannian manifold. Finally in Chapter 4 and 5 we introduce the work of Baudoin and Garofalo and apply their theory to two specific cases.

Riemannian manifolds with a Ricci lower bound is an important researched area of differential geometry, in particular in the context of comparison theorems. One result is a way to characterize Ricci curvature bounds through a condition of the Laplacian: the curvature-dimension inequality $\Gamma_2(f)\geq \frac{1}{n}(\Delta f)^2+\rho\Gamma(f)$ where $n$ is the dimension of the manifold, $\Gamma_2$ and $\Gamma$ two operators based on $\Delta$, the Laplacian and $\rho$ the bound of the Ricci curvature. Following this analytic, but equivalent, approach it is possible to find a generalization, which hing only on differential operator, also for non Riemannian cases where the Ricci curvature is not completely defined. In this thesis we study a generalization due to Baudoin and Garofalo for the case of sub-Riemannian manifolds with transverse symmetries. In Chapter 1 we set down the basic concepts we need throughout the thesis. Moreover in Chapter 2 we present the Riemannian curvature-dimension inequality and answer the question: what can be said about manifolds with a Ricci lower bound? Chapter 3 is entirely dedicated to introduce notions of sub-Riemannian manifold. Finally in Chapter 4 and 5 we introduce the work of Baudoin and Garofalo and apply their theory to two specific cases.

### A generalized Curvature-Dimension inequality in sub-Riemannian geometry

#### Abstract

Riemannian manifolds with a Ricci lower bound is an important researched area of differential geometry, in particular in the context of comparison theorems. One result is a way to characterize Ricci curvature bounds through a condition of the Laplacian: the curvature-dimension inequality $\Gamma_2(f)\geq \frac{1}{n}(\Delta f)^2+\rho\Gamma(f)$ where $n$ is the dimension of the manifold, $\Gamma_2$ and $\Gamma$ two operators based on $\Delta$, the Laplacian and $\rho$ the bound of the Ricci curvature. Following this analytic, but equivalent, approach it is possible to find a generalization, which hing only on differential operator, also for non Riemannian cases where the Ricci curvature is not completely defined. In this thesis we study a generalization due to Baudoin and Garofalo for the case of sub-Riemannian manifolds with transverse symmetries. In Chapter 1 we set down the basic concepts we need throughout the thesis. Moreover in Chapter 2 we present the Riemannian curvature-dimension inequality and answer the question: what can be said about manifolds with a Ricci lower bound? Chapter 3 is entirely dedicated to introduce notions of sub-Riemannian manifold. Finally in Chapter 4 and 5 we introduce the work of Baudoin and Garofalo and apply their theory to two specific cases.
##### Scheda Scheda DC
2021
A generalized Curvature-Dimension inequality in sub-Riemannian geometry
Riemannian manifolds with a Ricci lower bound is an important researched area of differential geometry, in particular in the context of comparison theorems. One result is a way to characterize Ricci curvature bounds through a condition of the Laplacian: the curvature-dimension inequality $\Gamma_2(f)\geq \frac{1}{n}(\Delta f)^2+\rho\Gamma(f)$ where $n$ is the dimension of the manifold, $\Gamma_2$ and $\Gamma$ two operators based on $\Delta$, the Laplacian and $\rho$ the bound of the Ricci curvature. Following this analytic, but equivalent, approach it is possible to find a generalization, which hing only on differential operator, also for non Riemannian cases where the Ricci curvature is not completely defined. In this thesis we study a generalization due to Baudoin and Garofalo for the case of sub-Riemannian manifolds with transverse symmetries. In Chapter 1 we set down the basic concepts we need throughout the thesis. Moreover in Chapter 2 we present the Riemannian curvature-dimension inequality and answer the question: what can be said about manifolds with a Ricci lower bound? Chapter 3 is entirely dedicated to introduce notions of sub-Riemannian manifold. Finally in Chapter 4 and 5 we introduce the work of Baudoin and Garofalo and apply their theory to two specific cases.
curvature dimension
sub Riemannian
geometry
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12608/42097