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Summary

In this chapter we introduce foliations and discuss some fundamental examples. We characterise the integrability of subbundles of tangent bundles in terms of both flatness and torsion-freeness of suitable affine connections. In the final section we discuss the simultaneous integrability of complementary distributions making up an almost product structure.

We introduce Bott connections in general, and we apply them to Lagrangian foliations in particular. This leads to a proof of Weinstein’s characterisation of affinely flat manifolds as leaves of Lagrangian foliations. We also prove a Darboux theorem for pairs consisting of a symplectic structure together with a Lagrangian foliation.

Let ${\mathcal {D}}$ and $T$ be, respectively, a $C^1$ distribution of $k$-planes and a normal $k$-current on ${\mathbb {R}}^n$. Then ${\mathcal {D}}$ has to be involutive at almost every superdensity point of the tangency set of $T$ with respect to ${\mathcal {D}}$.

Summary

In this chapter we collect some basic definitions ofdifferential geometry, in order to recall someuseful results and to fix the notation. We assumethe reader to be familiar with the definitions ofsmooth manifold and a smooth map betweenmanifolds.

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4 - Foliations and Connections

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Involutivity of distributions at points of superdense tangency with respect to normal currents

2 - Vector Fields

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3 results

4 - Foliations and Connections

Summary

Involutivity of distributions at points of superdense tangency with respect to normal currents

2 - Vector Fields

Summary