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Transactions of the American Mathematical Society

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The heat equation for Riemannian foliations

Authors: Seiki Nishikawa, Mohan Ramachandran and Philippe Tondeur
Journal: Trans. Amer. Math. Soc. 319 (1990), 619-630
MSC: Primary 58G11; Secondary 35K05, 53C12, 58A14
MathSciNet review: 987165
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Abstract: Let $ \mathcal{F}$ be a Riemannian foliation on a closed oriented manifold $ M$, with the transversal Laplacian $ {\Delta _B}$ acting on the basic forms $ \Omega _B^r(\mathcal{F})$ of degree $ r \geqslant 0$. We construct the fundamental solution $ e_B^r(x,y,t)$ for the basic heat operator $ \partial /\partial t + {\Delta _B}$, and prove existence and uniqueness for the solution of the heat equation on $ \Omega _B^r(\mathcal{F})$. As an application we give a new proof for the deRham-Hodge decomposition theorem for $ {\Delta _B}$ in $ \Omega _B^r(\mathcal{F})$, generalizing the approach to the classical deRham-Hodge theorem pioneered by Milgram and Rosenbloom.

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Keywords: Heat equation, Riemannian foliation
Article copyright: © Copyright 1990 American Mathematical Society