Generalized Laplace operator

Generalized Laplace operators are mathematical objects that are examined in differential geometry, especially in global analysis . The operators treated here are generalizations of the Laplace operator known from real analysis . These generalizations are necessary in order to be able to define the Laplace operator on a Riemannian manifold . These operators play an important role in the proofs for the Atiyah-Singer index theorem and the Atiyah-Bott fixed-point theorem .

definition

Let be an n-dimensional Riemannian manifold , a Hermitian vector bundle and a geometric differential operator of the second order. This is called the generalized Laplace operator, if for its main symbol ${\displaystyle (M,g)}$ ${\displaystyle \pi \colon E\to M}$ ${\displaystyle H\colon \Gamma ^{\infty }(M,E)\to \Gamma ^{\infty }(M,E)}$

${\displaystyle \sigma _{H}^{2}(x,\xi )=\|\xi \|^{2}}$

for and applies. The norm is induced by the Riemannian metric and therefore the definition is also dependent on the metric. ${\displaystyle x\in M}$ ${\displaystyle \xi \in T_{x}^{*}M}$

Examples

Some well-known examples of generalized Laplace operators are presented below. To this end, as in the definition, let us assume a -dimensional, compact Riemannian manifold and a vector bundle. ${\displaystyle (M,g)}$ ${\displaystyle n}$ ${\displaystyle \pi \colon E\to M}$

Laplace-Beltrami operator

definition

The Laplace-Beltrami operator is defined by

${\displaystyle \Delta f:=\operatorname {div} (\operatorname {grad} f).}$

for twice continuously differentiable functions . The gradient of the function denotes a vector field . The divergence on at the point is defined as the track of the linear mapping , wherein the Levi-Civita connection on is. If the domain of definition is not a real manifold but an open subset of the , the connection is the usual directional derivative and the divergence of a vector field known from real analysis. In this case the well-known Laplace operator is obtained. ${\displaystyle f\colon M\to \mathbb {R} }$ ${\displaystyle \operatorname {grad} f}$ ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle p\in M}$ ${\displaystyle \nabla X\colon T_{p}M\to T_{p}M}$ ${\displaystyle \xi \mapsto \nabla _{\xi }X}$ ${\displaystyle \nabla }$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \nabla }$ ${\displaystyle \operatorname {div} }$

Local coordinates

Let there be local coordinates on and the associated base fields of the tangential bundle. With for the components of the Riemann metric had referred this basis with respect. ${\displaystyle (x_{1},\dots ,x_{n})}$ ${\displaystyle M}$ ${\displaystyle {\tfrac {\partial }{\partial x_{1}}},\dots ,{\tfrac {\partial }{\partial x_{n}}}}$ ${\displaystyle g_{ij}}$ ${\displaystyle 1\leq i,j\leq n}$ ${\displaystyle g}$

The representation of the gradient in local coordinates is then ${\displaystyle \operatorname {grad} }$

${\displaystyle \operatorname {grad} f=\sum _{i,j}\left(g^{ij}{\frac {\partial f}{\partial x_{j}}}\right){\frac {\partial }{\partial x_{i}}}}$.

Here is the inverse matrix of the matrix . ${\displaystyle (g^{ij})}$ ${\displaystyle (g_{ij})}$

The representation of the divergence of a vector field is ${\displaystyle \textstyle X=\sum \limits _{i}X^{i}{\tfrac {\partial }{\partial x_{i}}}}$

${\displaystyle \operatorname {div} X={\frac {1}{\sqrt {\det g}}}\sum _{i}{\frac {\partial }{\partial x_{i}}}\left({\sqrt {\det g}}X^{i}\right)}$,

where is the determinant of the matrix . ${\displaystyle \det g}$ ${\displaystyle (g_{ij})}$

If you put these equations together, you get the local representation

${\displaystyle \Delta f=\operatorname {div} (\nabla f)={\frac {1}{\sqrt {\det g}}}\sum _{i,j}{\frac {\partial }{\partial x_{i}}}\left({\sqrt {\det g}}\,g^{ij}{\frac {\partial f}{\partial x_{j}}}\right)}$

of the Laplace-Beltrami operator with respect to the metric . If one uses the representation of the Euclidean metric tensor in polar , cylindrical or spherical coordinates for the Laplace-Beltrami operator , then one obtains the representation of the usual Laplace operator in these coordinate systems. ${\displaystyle g}$

Hodge-Laplace operator

Let be the space of the differential forms over and the outer derivative . The adjoint outer derivative is denoted by. Then the operator is called ${\displaystyle \textstyle {\mathcal {A}}(M):=\bigoplus _{i=1}^{n}{\mathcal {A}}^{i}(M)}$ ${\displaystyle M}$ ${\displaystyle \mathrm {d} :{\mathcal {A}}^{i}(M)\to {\mathcal {A}}^{i+1}(M)}$ ${\displaystyle \delta }$

${\displaystyle \Delta :=\mathrm {d} \delta +\delta \mathrm {d} =(\mathrm {d} +\delta )^{2}}$

Hodge-Laplace or Laplace-de-Rham operator and is a generalized Laplace operator. The names come from the fact that this operator is used in the classical Hodge theory and the closely related De Rham complex .

Dirac-Laplace operator

A Dirac operator

${\displaystyle D:\Gamma ^{\infty }(M,E)\to \Gamma ^{\infty }(M,E)}$

is defined in such a way that by squaring it induces a generalized Laplace operator. That is, is a generalized Laplace operator and is called a Dirac-Laplace operator . These Laplace operators play an important role in the proof of the index theorem. ${\displaystyle D^{2}:\Gamma ^{\infty }(M,E)\to \Gamma ^{\infty }(M,E)}$

Bochner-Laplace operator

definition

The Bochner-Laplace operator is defined with the metric relationship on the vector bundle . Furthermore, let the Levi-Civita connection and the connection induced by and on the bundle be ${\displaystyle \nabla ^{E}\colon \Gamma (M,E)\to \Gamma (T^{*}M\otimes E)}$ ${\displaystyle E}$ ${\displaystyle \nabla ^{T^{*}M}\colon \Gamma (M,T^{*}M)\to \Gamma (T^{*}M\otimes T^{*}M)}$ ${\displaystyle \nabla ^{T^{*}M\otimes E}}$ ${\displaystyle \nabla ^{E}}$ ${\displaystyle \nabla ^{T^{*}M}}$ ${\displaystyle T^{*}M\otimes E}$

then the Bochner-Laplace operator is through

${\displaystyle \Delta ^{E}\cdot :=-\operatorname {Tr} _{g}\left(\nabla ^{T^{*}M\otimes E}\nabla ^{E}\cdot \right)\,.}$

Are defined. The figure is the tensor taper with respect to the Riemannian metric. ${\displaystyle \operatorname {Tr} _{g}}$

An equivalent definition of the Bochner-Laplace operator is

${\displaystyle \Delta ^{E}:=-(\nabla ^{E})^{*}\nabla ^{E}.}$

It is the adjoint operator with respect to the Riemannian metric . ${\displaystyle (\nabla ^{E})^{*}}$ ${\displaystyle g}$

Local representation

Is chosen as the context, the Levi-Civita connection is obtained in the local coordinate with the orthonormal frame representation ${\displaystyle e_{1},\ldots ,e_{n}}$

${\displaystyle \Delta ^{E}=-\sum _{i=1}^{n}\left(\nabla _{e_{i}}^{E}\nabla _{e_{i}}^{E}-\nabla _{\nabla _{e_{i}}e_{i}}^{E}\right)\,.}$

properties

• A generalized Laplace operator is a geometric differential operator of order two.
• Since a generalized Laplace operator has the main symbol, as required in the definition , it is an elliptic differential operator . ${\displaystyle |\xi |^{2}}$
• Every second-order differential operator with positive definite principal symbol is a generalized Laplace operator with respect to a suitable Riemannian metric on the manifold and a suitable Hermitian metric on the vector bundle.
• If the cuts are smooth , then ${\displaystyle \phi ,\psi \in \Gamma ^{\infty }(M,E)}$

${\displaystyle g(\Delta ^{E}\phi ,\psi )=g(\nabla ^{E}\phi ,\nabla ^{E}\psi )}$.

• The operator is nonnegative and essentially self-adjoint with respect to . The definition of manifolds can be found in the article on density bundles . ${\displaystyle \Delta ^{E}}$ ${\displaystyle L^{2}(X,E)}$ ${\displaystyle L^{2}}$
• Every generalized Laplace operator uniquely determines a relationship on the vector bundle and a cut , so that where the Bochner-Laplace operator is. Every generalized Laplace operator agrees with the Bochner-Laplace operator except for a perturbation of order zero. ${\displaystyle H}$ ${\displaystyle \nabla ^{E}}$ ${\displaystyle E}$ ${\displaystyle B\in \Gamma ^{\infty }(M,\operatorname {End} (E))}$ ${\displaystyle H=\Delta ^{E}-B}$ ${\displaystyle \Delta ^{E}}$

swell

• Isaac Chavel: Eigenvalues ​​in Riemannian Geometry (= Pure and Applied Mathematics 115). Academic Press, Orlando FL et al. 1984, ISBN 0-12-170640-0 .
• Liviu I. Nicolaescu: Lectures on the geometry of manifolds. 2nd edition. World Scientific Pub Co., Singapore et al. 2007, ISBN 978-981-270853-3 .
• Martin Schottenloher: Geometry and Symmetry in Physics. Leitmotiv of Mathematical Physics (= Vieweg textbook Mathematical Physics ). Vieweg, Braunschweig et al. 1995, ISBN 3-528-06565-6 .

See also

• Vectorial Laplace operator

Individual evidence

1. ↑ Torsten Fließbach : General Theory of Relativity . 4th edition, Elsevier - Spektrum Akademischer Verlag, 2003, Chapter 17 Generalized vector operations ISBN 3-8274-1356-7
2. ^ HB Lawson, M. Michelsohn: Spin Geometry . Princeton University Press, 1989, ISBN 978-0691085425 , p. 123
3. ^ ^{A b} Nicole Berline, Ezra Getzler , Michèle Vergne : Heat kernels and Dirac operators (= basic teachings of mathematical sciences 298). Berlin et al. Springer 1992, ISBN 0-387-53340-0 , pp. 63-64.