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On a symplectic vector space $(V,\omega)$ with an inner product $g$, one can construct a canonical almost complex structure using polar decomposition. If $(M,\omega)$ is a symplectic manifold with a Riemannian metric $g$, one can use this construction to construct an almost complex structure $J_p$ on each tangent space $T_pM$ of $M$ for all $p\in M$. Why is this construction smooth?

This problem is equivalent to the following matrix problem. Suppose $B$ is a positive definite symmetric matrix which depends smoothly on a parameter $t$ in some open interval $(a,b)$. Then there exists a unique positive definite symmetric matrix $S$ such that $S(t)^2 = B(t)$ for $t\in (a,b)$. Why does $S$ depend smoothly on $t$?

Gilbert Easton
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  • For your matrix problem: that’s just the inverse function theorem for the smooth bijective submersion (hence diffeomorphism) $X \rightarrow X^2$ (from positive definite symmetric matrices to themselves), isn’t it? – Aphelli Mar 01 '21 at 20:41
  • I think you're right. Let $Z$ denote space of pos. def. symm. matrices. Let $\psi: Z\rightarrow Z$ be the map which sends $X\rightarrow X^2$. $\psi$ is a diffeo. Then $S = \psi^{-1}(B)$ is also smooth as the composition of two smooth functions. Sounds good. Thanks! – Gilbert Easton Mar 01 '21 at 22:44

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