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Even more important is the converse: as the next lemma shows, any such map that is multilinear over C ∞ (M ) defines a tensor field. 4. (Tensor Characterization Lemma) A map τ : T 1 (M ) × · · · × T 1 (M ) × T(M ) × · · · × T(M ) → C ∞ (M ) is induced by a kl -tensor field as above if and only if it is multilinear over C ∞ (M ). 1 if and only if it is ∞ multilinear over C (M ). 7. 4. 3 Definitions and Examples of Riemannian Metrics In this chapter we officially define Riemannian metrics and construct some of the elementary objects associated with them.

Show that F is a smooth tensor field if and only if whenever {Xi } are smooth vector fields and {ω j } are smooth 1-forms defined on an open set U ⊂ M , the function F (ω 1 , . . , ω l , X1 , . . , Xk ) on U , defined by F (ω 1 , . . , ω l , X1 , . . , Xk )(p) = Fp (ωp1 , . . , ωpl , X1 |p , . . , Xk |p ), is smooth. Tensor Bundles and Tensor Fields 21 An important property of tensor fields is that they are multilinear over the space of smooth functions. 6 shows that the function F (X1 , . . , Xk , ω 1 , .

If S is sufficiently far from being integrable, choosing a fiber metric on S results in a sub-Riemannian metric whose geometric properties closely reflect the complex-analytic properties of M as a subset of Cn . Another motivation for studying sub-Riemannian metrics arises from control theory. In this subject, one is given a manifold with a vector field depending on parameters called controls, with the goal being to vary the controls so as to obtain a solution curve with desired properties, often one that minimizes some function such as arc length.

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