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almost complex manifold
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Symplectic manifold, Symplectic manifold - Contact manifolds, Symplectic manifold - Lagrangian and other submanifolds, Symplectic manifold - Linear symplectic manifold, Symplectic manifold - Volume form, Kähler manifold, Poisson bracket, symplectic topology, symplectic vector space, almost complex manifold, symplectic group, symplectic matrix, tautological one-form

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almost complex manifold: Encyclopedia II - Kähler manifold - Definition

A Kähler metric on a complex manifold M is a hermitian metric on the complexified tangent bundle satisfying a condition that has several equivalent characterizations (the most geometric being that parallel transport gives rise to complex-linear mappings on the tangent spaces). In terms of local coordinates it is specified in this way: if is the hermitian metric, then the associated Kähler form (defined up to a factor of i/2) by is closed: that is, dω = 0. If M carries such a metric it is called a Kähler manifold. The metric on a Kähler ...

There are several natural geometric notions of submanifold of a symplectic manifold. There are symplectic submanifolds (potentially of any even dimension), where the symplectic form is required to induce a symplectic form on the submanifold. On isotropic submanifolds, the symplectic form restricts to zero, i.e. each tangent space is an isotropic subspace of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is coisotropic (the dual ...

almost complex manifold

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