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Adiabatic Limits and Foliations

Kefeng Liu·Weiping Zhang
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Abstract

We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes[Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with spin leaves, for what we call almost Riemannian foliations. Several new vanishing theorems are also proved by using our method.

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Kefeng Liu, Weiping Zhang. 1999-12-29. Adiabatic Limits and Foliations. https://arxiv.org/abs/math/9912223

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