Exploring Normal Bundle Over Totally Geodesic Submanifold Of Complete Manifold

Introduction

The fascinating realm of Riemannian geometry provides a rich tapestry of concepts, theorems, and problems that challenge and inspire mathematicians. Among these, the study of complete manifolds and their geodesic properties holds a place of paramount importance. This article delves into a specific problem within this domain, inspired by Problem 12-9 from Lee's Introduction to Riemannian Manifolds, which offers a compelling generalization of the celebrated Cartan-Hadamard theorem. We embark on an exploration of the normal bundle over a totally geodesic submanifold within a complete manifold, aiming to elucidate the conditions under which this normal bundle inherits the property of completeness. This investigation will not only shed light on the interplay between submanifolds and their ambient spaces but also underscore the profound implications of completeness in the context of Riemannian geometry. Riemannian geometry is a vast field, and understanding these concepts is crucial for further studies. The normal bundle plays a pivotal role in understanding the geometry of submanifolds.

Problem Statement and Background

At the heart of our discussion lies a generalization of the Cartan-Hadamard theorem. To fully appreciate the problem, let's first lay out the essential elements. Suppose we have a $(M, g)$ which is a complete, connected Riemannian manifold, and let $S$ be a closed, embedded submanifold of $M$. The question we seek to address is: under what conditions is the normal bundle $NS$ of $S$ also complete? This question naturally arises in the context of Riemannian geometry, where the notion of completeness plays a crucial role in determining the global structure of manifolds. Specifically, a Riemannian manifold is said to be complete if all geodesics can be extended indefinitely. Completeness is a fundamental property that ensures the manifold doesn't have any