Symplectic geometry of vector bundle maps of tangent bundles (Q1596560)

Suppose \((M,g)\) is a Riemannian manifold. Then \(TM\) is equipped with a Sasaki metric \(\widehat{g}\). There is a \(\widehat{g}\)-orthogonal decomposition \(TTM=\mathcal{H}\oplus \mathcal{V}\) into horizontal and vertical subbundles. An almost-complex structure \(J\) for \(TM\) compatible with \(\widehat{g}\) is defined as follows: \(J(X_\xi^H+Y_\xi^V)=X_\xi^V-Y_\xi^H\). The 2-form \(\omega=\widehat{g}(J*,*)\) is exactly \(D^*(\omega_c)\) where \(D:TM\to T^*M\) is the dual map induced by \(g\) and \(\omega_c\) is the canonical symplectic form on \(T^*M\). The triple \((J,\widehat{g},\omega)\) is called the canonical almost Kähler structure of \(TM\). Suppose \(F:TM \to TM'\) is a map between the tangent bundles of two Riemannian manifolds over a map \(f:(M,g)\to (M',g')\). Then the map \(F\) is called \(H\)-isotropic if \(F_*(\mathcal{H}_\xi)\) is an isotropic subspace of \(T_{F(\xi)}TM'\) with respect to \(\omega'\) for all \(\xi\in TM\), where \(H_\xi=(\mathcal{H}TM)_\xi\). In this article the author studies properties of the \(H\)-isotropic maps. Various conditions on maps \(f\) are obtained, from which follows that \(f_*\) is is \(H\)-isotropic. Theorem 1. Suppose \(\text{dim} M\geq 2\) and \(f:(M,g)\to (M',g')\) is a map of Riemannian manifolds. Suppose there exists a function \(c(x)>0\) on \(M\) such that \(f^*g'=cg\). Let \(F=\frac {c_1}{c}f_*\) for some real-valued function \(c_1\) on \(M\). Then \(F\) is \(H\)-isotropic if and only if \(c_1=const\). In particular, for every nonzero constant \(c_2\), the map \(\frac {c_2}{c}f_*\) is symplectically homothetic. Theorem 2. Suppose \(f:(M,g)\to (M',g')\) and \(\text{dim} M\geq 2\). (i)\ If \(f_*\) is \(H\)-isotropic, and, for every \(x\in M\), there exists \(\xi(x)\in T_xM\) and \(c(x)\neq 0\) such that \((f_*)^*(\omega_{f_*(\xi(x))}') =c(x)\omega_{\xi(x)}\), then \(f_*\) is symplectically homothetic. (ii)\ \(f_*\) is symplectically conformal \(\Longleftrightarrow\) \(f_*\) is symplectically homothetic \(\Longleftrightarrow\) \(f\) is homothetic \(\Longleftrightarrow\) \(f\) is conformal and \(f_*\) is \(H\)-isotropic.