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- Bismut_connection abstract "In mathematics, the Bismut connection is the unique connection on a complex manifold that satisfies the following conditions, It preserves the metric It preserves the complex structure The torsion contracted with the metric, i.e. , is totally skew-symmetric. Bismut has used this connection when proving a local index formula for the Dolbeault operator on non-Kähler manifolds. Bismut connection has applications in type II and heterotic string theory.The explicit construction goes as follows. Let denote the pairng of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. . Further let be the Levi-Civita connection. Define first a tensor such that . It is easy to see that this tensor is anti-symmetric in the first and last entry, i.e. the new connection still preserves the metric. In concrete terms, the new connection is given by with being the Levi-Civita connection. It is also easy to see that the new connection preserves the complex structure. However, the tensor is not yet totall anti-symmetric, in fact the anti-symmetrization will lead to the Nijenhuis tensor. Denote the anti-symmetrization as , with given explicitly asWe show that still preserves the complex structure (that it preserves the metric is easy to see), i.e. .So if is integrable, then above term vanishes, and the connectiongives the Bismut connection.".
- Bismut_connection wikiPageID "19454484".
- Bismut_connection wikiPageRevisionID "601815878".
- Bismut_connection hasPhotoCollection Bismut_connection.
- Bismut_connection subject Category:Complex_manifolds.
- Bismut_connection type Artifact100021939.
- Bismut_connection type ComplexManifolds.
- Bismut_connection type Conduit103089014.
- Bismut_connection type Manifold103717750.
- Bismut_connection type Object100002684.
- Bismut_connection type Passage103895293.
- Bismut_connection type PhysicalEntity100001930.
- Bismut_connection type Pipe103944672.
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- Bismut_connection type Way104564698.
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- Bismut_connection comment "In mathematics, the Bismut connection is the unique connection on a complex manifold that satisfies the following conditions, It preserves the metric It preserves the complex structure The torsion contracted with the metric, i.e. , is totally skew-symmetric. Bismut has used this connection when proving a local index formula for the Dolbeault operator on non-Kähler manifolds. Bismut connection has applications in type II and heterotic string theory.The explicit construction goes as follows.".
- Bismut_connection label "Bismut connection".
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- Bismut_connection wasDerivedFrom Bismut_connection?oldid=601815878.
- Bismut_connection isPrimaryTopicOf Bismut_connection.