An Almost-Kähler Tangent Bundle using the Sasaki Metric, Canonical Sympletic Form (Pulled-Back to the Tangent Bundle), and Compatible Triple Almost-Complex Structure is Kähler if and only if (Q,g) is Flat

Consider a Rienmannian manifold \((Q,g)\). The Sasaki metric \(g^S\) is the canonical lift of of \(g\) to the tangent bundle of \(Q\) https://g.co/gemini/share/7ad77f487a10. Note that \(\det(g^S) = (\det(g))^2\).

\(T^*(Q)\) naturally has a canonical sympletic form \(\omega\) on it. Using the musical isomorphism \(\flat: TQ \to T^*(Q)\) induced by \(g^S\), we may pullback \(\omega\) to be a symplectic form \(\Omega = \flat^*(\omega)\) on \(TQ\) https://g.co/gemini/share/551038e10dd7. Note that \(\det(\Omega) = (\det(g))^2\).

We may then complete the compatible triple with an almost complex structure \(J\) on \(TQ\), having \(u^T\Omega v = u^T g^S Jv\) and  \(J^2 = -I\ \in \mathbb{R}^{2n \times 2n}\) in sympletic coordinates, creating a natural almost-Kähler structure on \(TQ\).  In sympletic coordinates, one does a \(g^S\)-polar decomposition on \(A =  (g^S)^{-1}\Omega\). This is accomplished by forming the \(g^S\)-symmetrization of \(A\), \(P\text{Sq} = A^Tg^SA\), finding a symmetric matrix \(P\) with \(Pg^SP = P\text{Sq}\) and \(u^Tg^Su >0\) for all eigenvectors \(u\) of \(P\), then setting \(J = AP^{-1}\). In practice, one finds \(P\) by performing a Cholesky decomposition on \(g^S\), writing \(g^S = Z^TZ\), setting \(A_{\text{std}} = ZAZ^T\), computing \(P_{\text{std}} = \sqrt{A_{\text{std}}}\), then \(P = Z^{-1}P_{\text{std}}Z^{-T}\). Note that \(\det(P) = \det(J) = 1\) in this case.

(To be shown later: \(TQ\) with \((g^S,\Omega,J)\) on it is Kähler if and only if \((Q,g)\) is flat https://gemini.google.com/u/1/app/7824cf2e6f205c05. It has a vanishing first Chern class. It would then remain to check that \((TQ, (g^S))\) has vanishing Ricci curvature, making \((TQ, (g^S, \Omega, J))\) non-compact Calabi-Yau.)

(This link gives hope that for some flat, orientable 3-manifolds \((Q, g)\) that are mapping tori of a \(T^2\), one should also have that \((TQ, (g^S, \Omega, J))\) is not only Kähler but has a vanishing first Chern class https://gemini.google.com/u/1/app/dce555be078681d6.We may then hope that \((TQ, g^S)\) has vanishing Ricci curvature. We'll have to see if I can design some mechanisms that have these mapping tori as thei c-spaces. #MoreOfAPromptEngineerThanAMathematician #Bazinga)

The following is a MATLAB script to compute the compatible triple on the tangent bundle for Riemannian configuration space (c-space) of the 2D robot arm outlined in this webpage: compatibleTriple.m. (You will need the robotBuilder.m script to run the compatibleTriple.m script.)
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