The subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume and length given by Bishop-Gromov and Toponogov. An application is Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. Using convexity arguments, we prove the ”soul theorem” of Cheeger and Gromoll and the sphere theorem of Berger and Klingenberg for nonnegative curvature. If lower Ricci curvature bounds are given we exploit subharmonicity instead of convexity and show the rigidity theorems of Myers-Cheng and the splitting theorem of Cheeger and Gromoll. The Bishop-Gromov inequality shows polynomial growth of finitely generated subgroups of the fundamental group of a space with nonnegative Ricci curvature (Milnor). We also discuss briefly Bochner’s method. The leading principle of the whole exposition is the use of convexity methods. Five ideas make these methods work: The comparison theory for the Riccati ODE, which probably goes back to L.Green [15] and which was used more systematically by Gromov [20], the triangle inequality for the Riemannian distance, the method of support function by Greene and Wu [16],[17],[34], the maximum principle of E.Hopf, generalized by E.Calabi [23], [4], and the idea of critical points of the distance function which was first used by Grove and Shiohama [21]. We have tried to present the ideas completely without being too technical. These notes are based on a course which I gave at the University of Trento in March 1994. It is a pleasure to thank Elisabetta Ossanna and Stefano Bonaccorsi who have worked out and typed part of these lectures. We also thank Evi Samiou and Robert Bock for many valuable corrections.

Comparison Theorems in Riemannian Geometry

This paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part, subject to Dirichlet or Neumann boundary controls. The observability inequalities are established by the Riemannian geometry method under some geometric condition for the Dirichlet problem and for the Neumann problem, respectively. Next, a number of nontrivial examples are presented to verify the observability inequality. In particular, a counterexample is given without boundary exact controllability, where the control is exerted on the whole boundary.

On The Observability Inequalities for Exact Controllability of Wave Equations With Variable Coefficients

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.

The inverse problem for the local geodesic ray transform

/pdf/c-infty-approximations-of-convex-subharmonic-and-2qmvpxwhg7.pdf

$C^\infty $ approximations of convex, subharmonic, and plurisubharmonic functions

We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.

Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds

/pdf/c-convex-functions-and-manifolds-of-positive-curvature-1bs78x7tme.pdf

C∞ convex functions and manifolds of positive curvature

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

R. E. Greene

Author Tools

Chat about Author

Papers

C∞ convex functions and manifolds of positive curvature