Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each tangent space. One can also define a Hermitian manifold as a complex manifold with a Riemannian metric that preserves the almost complex structure J. There are several related concepts coming from integrability conditions: A Hermitian manifold has a unitary structure, with the almost complex structure integrable. Without the integrability condition, the notion is an almost Hermitian manifold. With an additional integrability condition on the Sp- structure, one obtains a Kahler manifold. With integrability on the Sp-structure but not the almost complex structure, one obtains an almost Kahler manifold

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