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Infinitesimal aspects of the Laplace operator

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Anders Kock

In the context of synthetic differential
geometry, we study the Laplace operator an a Riemannian
manifold. The main new aspect is a neighbourhood of the
diagonal, smaller than the second neighbourhood usually
required as support for second order differential operators.
The new neighbourhood has the property that a function is affine on it
if and only if it is harmonic.

Keywords: Laplacian, harmonic, conformal, synthetic dfferential geometry.

2000 MSC: 18F99, 53B20.

*Theory and Applications of Categories*, Vol. 9, 2001, No. 1, pp 1-16.

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