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• Print publication year: 2016
• Online publication date: July 2016

# 2 - Geometrical tools

## Summary

We can regard a compact smooth manifold as built up by glueing together smaller pieces, which are easier to analyse. In this chapter we begin the description of this process. After obtaining some basic results on Riemannian metrics, we study geodesics for such metrics. The key result is that any two nearby points are joined by a unique shortest geodesic. This leads us to study the way in which a closed submanifold lies in a manifold: we describe the structure of a neighbourhood of the submanifold as having the form of a tube.

A diffeotopy, or differentiable isotopy, can be considered either as deforming the embedding of one manifold in another or as an embedding of a product with I. If the deformation can be extended to the whole manifold, the two embeddings are equivalent. The diffeotopy extension theorem asserts that under certain conditions, this extension is possible; it may thus be looked on as a uniqueness theorem. We apply this result to obtain a uniqueness theorem for tubular neighbourhoods, which enables us to pass from knowledge of the structure of a compact submanifold M of a manifold N to knowledge of a neighbourhood of M: the only extra piece of information needed is the structure of the normal bundle. This contributes to the general aim of building up global results from merely local ones.

We define inverse procedures for straightening a corner, to yield a manifold with boundary, and for introducing corners: it will be useful in Chapter 5 to be able to effectively ignore corners.

Finally we discuss glueing and the inverse process of cutting: these are simple geometrical constructions which, given some smooth manifolds (perhaps with boundaries and corners) and additional data where necessary, give rise to new manifolds. On account of their perspicuity, these methods are traditional in describing the topology of surfaces, and they remain a very powerful tool in higher dimensions.

Riemannian metrics

We recall that if Mm is a smooth manifold, the bundle over M associated to the tangent bundle and whose fibre over P is the set of all positive definite quadratic forms on TPM is called the Riemann bundle, and any cross-section of it a Riemannian structure on M; in local coordinates this takes the form.

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