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Gilkey, P. B.; Park, J. H.

The Bochner Laplacian, Riemannian submersions, heat content asymptotics, and heat equation asymptotics. (English) Zbl 0955.53022

Czech. Math. J. 49, No. 2, 233-240 (1999).
Let \(\pi : Z \rightarrow Y\) be a Riemannian submersion where the fibers of \(\pi \) are compact and where \(Y\) is a compact manifold with smooth boundary. The main result of the paper reads as follows. Let \(\Delta ^0_Y\) or \(\Delta ^0_Z\) be the usual scalar Laplacian of \(Y\) or of \(Z\), respectively. Then \(\pi ^*\Delta ^0_Y = \Delta ^0_Z \pi ^*\) if and only if the fibers of \(\pi \) are minimal submanifolds of \(Z\). Moreover, some formulae for the heat kernel of the corresponding Bochner Laplacians are deduced.
Reviewer: I.Kolář (Brno)

MSC:

53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

Riemannian submersion; minimal submanifold; Bochner Laplacian; heat equation
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\textit{P. B. Gilkey} and \textit{J. H. Park}, Czech. Math. J. 49, No. 2, 233--240 (1999; Zbl 0955.53022)
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