synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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The magic algebraic facts
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differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
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(..)
Given a Riemannian manifold , the Laplace-Beltrami operator is the differential operator on the space of smooth functions given by the formula
where
is the de Rham differential (depending only on the smooth manifold , not on the metric tensor);
is the Hodge star operator of (this is where the dependence on the metric tensor enters).
The same formula makes sense more generally for pseudo-Riemannian manifolds. Even so, in the pseudo-Riemannian case one tends to speak of the wave operator instead of the Laplace operator, and to use the symbol instead of (at least for flat pseudo-Riemannian manifolds: Minkowski spacetime).
(coordinate-expression of Laplace operator)
If is a chart of with coordinate functions , then the Laplace operator (1) is equivalently given by the following component-expression:
Here:
is the signature of the metric tensor, hence
denotes the partial derivative by the coordinate ;
is the square matrix of components of the metric tensor in the given coordinate chart, hence such that
is the determinant of this matrix,
its absolute value (taking this is irrelevant for Riemannian manifolds but necessary for pseudo-Riemannian manifolds, where the determinan of the metric is negative),
is the positive square root of that;
is the corresponding inverse matrix;
the Einstein summation convention is used throughout, meaing that a sum over repeated indices is understood.
Using
the component formula for the de Rham differential ,
the component formula for the Hodge star operator (see there)
we compute as follows:
The functional determinant of Laplace operator on a given space of differential p-forms appears as factor of the analytic torsion of the given Riemannian manifold.
Textbook accounts:
See also:
Wikipedia, Laplace operator, Laplace-Beltrami operator
MathOverflow: why-is-the-laplacian-ubiquitous
Last revised on August 23, 2024 at 06:39:50. See the history of this page for a list of all contributions to it.