209 research outputs found
A geometric theory of harmonic and semi-conformal maps
We describe for any Riemannian manifold a certain infinitesimal neighbourhood
of the diagonal. Semi-conformal maps are analyzed as those that preserve such
neighbourhoods; harmonic maps are analyzed as those that preserve mirror image
formation for pairs of points in such neighbourhoods.Comment: 20 page
Monads and extensive quantities
If T is a commutative monad on a cartesian closed category, then there exists
a natural T-bilinear pairing from T(X) times the space of T(1)-valued functions
on X ("integration"), as well as a natural T-bilinear action on T(X) by the
space of these functions. These data together make the endofunctors T and
"functions into T(1)" into a system of extensive/intensive quantities, in the
sense of Lawvere. A natural monad map from T to a certain monad of
distributions (in the sense of functional analysis (Schwartz)) arises from this
integration
Infinitesimal aspects of the Laplace operator
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.Comment: 19 page
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