Difference between revisions of "Hermitian metric"
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− | A Hermitian metric on a complex vector | + | A Hermitian metric on a complex vector space $ V $ |
+ | is a positive-definite [[Hermitian form|Hermitian form]] on $ V $. | ||
+ | The space $ V $ | ||
+ | endowed with a Hermitian metric is called a unitary (or complex-Euclidean or Hermitian) vector space, and the Hermitian metric on it is called a Hermitian scalar product. Any two Hermitian metrics on $ V $ | ||
+ | can be transferred into each other by an automorphism of $ V $. | ||
+ | Thus, the set of all Hermitian metrics on $ V $ | ||
+ | is a homogeneous space for the group $ \mathop{\rm GL} _ {n} ( \mathbf C ) $ | ||
+ | and can be identified with $ \mathop{\rm GL} _ {n} ( \mathbf C ) / U ( n) $, | ||
+ | where $ n = \mathop{\rm dim} V $. | ||
− | Every complex vector bundle has a Hermitian metric. A [[Connection|connection]] | + | A complex vector space $ V $ |
+ | can be viewed as a real vector space $ V ^ {\mathbf R} $ | ||
+ | endowed with the operator defined by the complex structure $ J ( x) = ix $. | ||
+ | If $ h $ | ||
+ | is a Hermitian metric on $ V $, | ||
+ | then the form $ g = \mathop{\rm Re} h $ | ||
+ | is a Euclidean metric (a scalar product) on $ V $ | ||
+ | and $ \omega = \mathop{\rm Im} h $ | ||
+ | is a non-degenerate skew-symmetric bilinear form on $ V $. | ||
+ | Here $ g ( Jx, Jy ) = g ( x , y) $, | ||
+ | $ \omega ( Jx , Jy) = \omega ( x , y) $ | ||
+ | and $ \omega ( x , y) = g ( x , Jy) $. | ||
+ | Any of the forms $ g $, | ||
+ | $ \omega $ | ||
+ | determines $ h $ | ||
+ | uniquely. | ||
+ | |||
+ | A Hermitian metric on a complex vector bundle $ \pi : E \rightarrow M $ | ||
+ | is a function $ g : p \mapsto g _ {p} $ | ||
+ | on the base $ M $ | ||
+ | that associates with a point $ p \in M $ | ||
+ | a Hermitian metric $ g _ {p} $ | ||
+ | in the fibre $ E ( p) = \pi ^ {-} 1 ( p) $ | ||
+ | of $ \pi $ | ||
+ | and that satisfies the following smoothness condition: For any smooth local sections $ e $ | ||
+ | and $ e ^ {*} $ | ||
+ | of $ \pi $ | ||
+ | the function $ p \mapsto g _ {p} ( e _ {p} , e _ {p} ^ {*} ) $ | ||
+ | is smooth. | ||
+ | |||
+ | Every complex vector bundle has a Hermitian metric. A [[Connection|connection]] $ \nabla $ | ||
+ | on a complex vector bundle $ \pi $ | ||
+ | is said to be compatible with a Hermitian metric $ g $ | ||
+ | if $ g $ | ||
+ | and the operator $ J $ | ||
+ | defined by the complex structure in the fibres of $ \pi $ | ||
+ | are parallel with respect to $ \nabla $( | ||
+ | that is, $ \nabla g = \nabla J = 0 $), | ||
+ | in other words, if the corresponding parallel displacement of the fibres of $ \pi $ | ||
+ | along curves on the base is an isometry of the fibres as unitary spaces. For every Hermitian metric there is a connection compatible with it, but the latter is, generally speaking, not unique. When $ \pi $ | ||
+ | is a holomorphic vector bundle over a complex manifold $ M $( | ||
+ | see [[Vector bundle, analytic|Vector bundle, analytic]]), there is a unique connection $ \nabla $ | ||
+ | of $ \pi $ | ||
+ | that is compatible with a given Hermitian metric and that satisfies the following condition: The covariant derivative of any holomorphic section $ e $ | ||
+ | of $ \pi $ | ||
+ | relative to any anti-holomorphic complex vector field $ \overline{X}\; $ | ||
+ | on $ M $ | ||
+ | vanishes (the canonical Hermitian connection). The [[Curvature|curvature]] form of this connection can be regarded as a $ 2 $- | ||
+ | form of type $ ( 1 , 1 ) $ | ||
+ | on $ M $ | ||
+ | with values in the bundle of endomorphisms of $ \pi $. | ||
+ | The canonical connection can also be viewed as a connection on the principal $ \mathop{\rm GL} _ {n} ( \mathbf C ) $- | ||
+ | bundle $ \widetilde \pi : P \rightarrow M $ | ||
+ | associated with the holomorphic vector bundle $ \pi $ | ||
+ | of complex dimension $ n $. | ||
+ | It can be characterized as the only connection on $ \widetilde \pi $ | ||
+ | with complex horizontal subspaces in the tangent spaces of the complex manifold $ P $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''2''' , Interscience (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''2''' , Interscience (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
A complex vector bundle on which a Hermitian metric is given is called a Hermitian vector bundle. | A complex vector bundle on which a Hermitian metric is given is called a Hermitian vector bundle. |
Revision as of 22:10, 5 June 2020
A Hermitian metric on a complex vector space $ V $
is a positive-definite Hermitian form on $ V $.
The space $ V $
endowed with a Hermitian metric is called a unitary (or complex-Euclidean or Hermitian) vector space, and the Hermitian metric on it is called a Hermitian scalar product. Any two Hermitian metrics on $ V $
can be transferred into each other by an automorphism of $ V $.
Thus, the set of all Hermitian metrics on $ V $
is a homogeneous space for the group $ \mathop{\rm GL} _ {n} ( \mathbf C ) $
and can be identified with $ \mathop{\rm GL} _ {n} ( \mathbf C ) / U ( n) $,
where $ n = \mathop{\rm dim} V $.
A complex vector space $ V $ can be viewed as a real vector space $ V ^ {\mathbf R} $ endowed with the operator defined by the complex structure $ J ( x) = ix $. If $ h $ is a Hermitian metric on $ V $, then the form $ g = \mathop{\rm Re} h $ is a Euclidean metric (a scalar product) on $ V $ and $ \omega = \mathop{\rm Im} h $ is a non-degenerate skew-symmetric bilinear form on $ V $. Here $ g ( Jx, Jy ) = g ( x , y) $, $ \omega ( Jx , Jy) = \omega ( x , y) $ and $ \omega ( x , y) = g ( x , Jy) $. Any of the forms $ g $, $ \omega $ determines $ h $ uniquely.
A Hermitian metric on a complex vector bundle $ \pi : E \rightarrow M $ is a function $ g : p \mapsto g _ {p} $ on the base $ M $ that associates with a point $ p \in M $ a Hermitian metric $ g _ {p} $ in the fibre $ E ( p) = \pi ^ {-} 1 ( p) $ of $ \pi $ and that satisfies the following smoothness condition: For any smooth local sections $ e $ and $ e ^ {*} $ of $ \pi $ the function $ p \mapsto g _ {p} ( e _ {p} , e _ {p} ^ {*} ) $ is smooth.
Every complex vector bundle has a Hermitian metric. A connection $ \nabla $ on a complex vector bundle $ \pi $ is said to be compatible with a Hermitian metric $ g $ if $ g $ and the operator $ J $ defined by the complex structure in the fibres of $ \pi $ are parallel with respect to $ \nabla $( that is, $ \nabla g = \nabla J = 0 $), in other words, if the corresponding parallel displacement of the fibres of $ \pi $ along curves on the base is an isometry of the fibres as unitary spaces. For every Hermitian metric there is a connection compatible with it, but the latter is, generally speaking, not unique. When $ \pi $ is a holomorphic vector bundle over a complex manifold $ M $( see Vector bundle, analytic), there is a unique connection $ \nabla $ of $ \pi $ that is compatible with a given Hermitian metric and that satisfies the following condition: The covariant derivative of any holomorphic section $ e $ of $ \pi $ relative to any anti-holomorphic complex vector field $ \overline{X}\; $ on $ M $ vanishes (the canonical Hermitian connection). The curvature form of this connection can be regarded as a $ 2 $- form of type $ ( 1 , 1 ) $ on $ M $ with values in the bundle of endomorphisms of $ \pi $. The canonical connection can also be viewed as a connection on the principal $ \mathop{\rm GL} _ {n} ( \mathbf C ) $- bundle $ \widetilde \pi : P \rightarrow M $ associated with the holomorphic vector bundle $ \pi $ of complex dimension $ n $. It can be characterized as the only connection on $ \widetilde \pi $ with complex horizontal subspaces in the tangent spaces of the complex manifold $ P $.
References
[1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969) |
[2] | A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French) |
[3] | R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) |
[4] | A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958) |
Comments
A complex vector bundle on which a Hermitian metric is given is called a Hermitian vector bundle.
Hermitian metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_metric&oldid=18912