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Hermitian Metric


A Hermitian metric on a complex vector bundle assigns a Hermitian inner product to every fiber bundle. The basic example is the trivial bundle pi:U×C^k->U, where U is an open set in R^n. Then a positive definite Hermitian matrix H defines a Hermitian metric by

 <v,w>=v^(T)Hw^_,

where w^_ is the complex conjugate of w. By a partition of unity, any complex vector bundle has a Hermitian metric.

In the special case of a complex manifold, the complexified tangent bundle TM tensor C may have a Hermitian metric, in which case its real part is a Riemannian metric and its imaginary part is a nondegenerate alternating multilinear form omega. When omega is closed, i.e., in this case a symplectic form, then omega is a Kähler form.

On a holomorphic vector bundle with a Hermitian metric h, there is a unique connection compatible with h and the complex structure. Namely, it must be del =partial+partial^_, where partials=h^(-1)partialhs in a trivialization.


See also

Complex Geometry, Complex Manifold, Complex Vector Bundle, Holomorphic Vector Bundle, Kähler Form, Kähler Manifold, Riemannian Metric, Symplectic Form, Unitary Group

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Hermitian Metric." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HermitianMetric.html

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