A Canonical Compatible Metric for Geometric Structures on by Lauret J.

By Lauret J.

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Wilson proved that (N , ·, · ) and (N , ·, · ) are isometric if and only if ·, · = ϕ. ·, · := ϕ −1 ·, ϕ −1 · for some ϕ ∈ Aut(n) (see the proof of [30, Theorem 3]). Therefore, although the Lie bracket μ does not play any role in the definition of a compatible metric, it is crucial in the study of the moduli space of compatible metrics on (N , γ ) up to isometry. 38]), ric ·,· (X, Y ) = Ric ·,· X, Y = − + 1 2 1 4 μ(X, X i ), X j μ(Y, X i ), X j ij μ(X i , X j ), X μ(X i , X j ), Y , (28) ij 137 A CANONICAL COMPATIBLE METRIC for all X, Y ∈ n, where {X 1 , .

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