Book contents
- Frontmatter
- Contents
- Preface
- 0 The background: vector calculus
- 1 Affine spaces
- 2 Curves, functions and derivatives
- 3 Vector fields and flows
- 4 Volumes and subspaces: exterior algebra
- 5 Calculus of forms
- 6 Frobenius's theorem
- 7 Metrics on affine spaces
- 8 Isometries
- 9 Geometry of surfaces
- 10 Manifolds
- 11 Connections
- 12 Lie groups
- 13 The tangent and cotangent bundles
- 14 Fibre bundles
- 15 Connections revisited
- Bibliography
- Index
6 - Frobenius's theorem
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- 0 The background: vector calculus
- 1 Affine spaces
- 2 Curves, functions and derivatives
- 3 Vector fields and flows
- 4 Volumes and subspaces: exterior algebra
- 5 Calculus of forms
- 6 Frobenius's theorem
- 7 Metrics on affine spaces
- 8 Isometries
- 9 Geometry of surfaces
- 10 Manifolds
- 11 Connections
- 12 Lie groups
- 13 The tangent and cotangent bundles
- 14 Fibre bundles
- 15 Connections revisited
- Bibliography
- Index
Summary
If ϕ:B → A is a smooth map of affine spaces then, for any y ∈ B, the set of vectors {ϕ*w | w ∈ TyB} is a linear subspace of Tϕ(y)A. It would be natural to think of this vector subspace as consisting of those vectors in Tϕ(y)A which are tangent to the image ϕ(B) of B under ϕ. In general this idea presents difficulties, which will be explained in later chapters; but one case of particular interest, in which the notion is a sensible one, arises when ϕ*y is an injective map for all y ∈ B, so that the space {ϕ*w | w ∈ TyB} has the same dimension as B for all y. In this case we call the image ϕ(B) a submanifold of A (this terminology anticipates developments in Chapter 10 and is used somewhat informally in the present chapter). Since it has an m-dimensional tangent space at each point (where m = dim B) the submanifold ϕ(B) is regarded as an m-dimensional object. Our assumption of injectivity entails that m ≤ n = dim A.
A curve (other than one which degenerates to a point) defines a submanifold of dimension 1, the injectivity of the tangent map corresponding in this case to the assumption that the tangent vector to the curve never vanishes. We regard R, for this purpose, as a 1-dimensional affine space.
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- Applicable Differential Geometry , pp. 140 - 163Publisher: Cambridge University PressPrint publication year: 1987