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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.
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 as
We 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 connection
gives the Bismut connection.
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