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  • Print publication year: 1987
  • Online publication date: August 2012

6 - Frobenius's theorem

Summary

If ϕ:BA is a smooth map of affine spaces then, for any yB, the set of vectors {ϕ*w | wTyB} 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 yB, so that the space {ϕ*w | wTyB} 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 mn = 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.