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.