A Hermitian manifold is a complex manifold whose tangent bundle carries a smoothly varying positive-definite Hermitian form. Its real part is a Riemannian metric invariant under the complex structure .
For the underlying Riemannian metric and complex structure , the fundamental form is . It is a real (1, 1)-form. In complex dimension , the compatible orientation has Riemannian volume form , and its Hodge star operator satisfies
(1)
A differential form of type (p, q) of type is real when . Locally this is equivalent to
(1)
for a Hermitian matrix .
A real -form is positive when
(1)
for every nonzero tangent vector of type . Equivalently, the local Hermitian matrix in the representation is positive definite.
If and are positive real -forms and are linearly independent tangent vectors of type , then
(1)
After simultaneous diagonalization by congruence, with positive diagonal entries , the left side is
(2)
which is positive because some minor is nonzero.

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  1. Complex manifold
  2. Integrable almost complex structure
  3. Almost complex manifold
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Hermitian manifold by Wikipedia Bot 1
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A Hermitian manifold is a type of complex manifold equipped with a Riemannian metric that is compatible with the complex structure. More formally, a Hermitian manifold consists of the following components: 1. **Complex Manifold**: A manifold \( M \) that is equipped with an atlas of charts where the transition functions are holomorphic mappings. This means that the local coordinates can be expressed in terms of complex variables.
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