Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 16 / 6 / a

Table of contents
- Solution a

Solution

An almost complex structure is a smooth bundle endomorphism

J : TM \to TM

with

J^{2} = - I

. It is an

ω

-compatible almost complex structure when

ω (J v, J w) = ω (v, w), ω (v, J v) > 0 (v \neq = 0) .

These conditions make

g_{J} (v, w) = ω (v, J w)

a Riemannian metric. In particular it is symmetric: invariance and

J^{2} = - I

give

ω (J v, w) = - ω (v, J w)

, and skew-symmetry then gives

g_{J} (w, v) = g_{J} (v, w)

. Positivity is the second condition. Moreover

J

is an isometry for

g_{J}

. Compatibility does not require that the almost complex structure be integrable.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version