 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-16/6/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

J : TM \to TM

with

J^{2} = - I

. It is an

ω

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.

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