Tangentially positive isometric actions and conjugate points

Let $\left({\mathrm{M}}, g\right)$ be a complete Riemannian manifold with no conjugate points and $f\colon \left({\mathrm{M}}, g\right) \to \left({\mathrm{B}}, g_{\mathrm{B}}\right)$ a principal $G$-bundle, where $G$ is a Lie group acting by isometries and ${\mathrm{B}}$ the smooth quotient with $g_{\mathrm{B}}$ the Riemannian submersion metric.

We obtain a characterization of conjugate point-free quotients $\left({\mathrm{B}}, g_{\mathrm{B}}\right)$ in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle $T{\mathrm{M}}$, from which we then derive necessary conditions, involving $G$ and ${\mathrm{M}}$, for the quotient metric to be conjugate point-free, particularly for ${\mathrm{M}}$ a reducible Riemannian manifold.

Let ${\mu_G}\colon T{\mathrm{M}}\to {\mathfrak{G}}^*$, with ${\mathfrak{G}}$ the Lie Algebra of $G$, be the moment map of the tangential $G$-action on $T{\mathrm{M}}$ and let ${\mathbf{G}}_{\mathbf {P}}$ be the canonical pseudo-Riemannian metric on $T{\mathrm{M}}$ defined by the symplectic form $d\Theta$ and the map $F\colon T{\mathrm{M}}\to {\mathrm{M}}\times {\mathrm{M}}$, $F(z)=\left( \exp(-z), \exp(z)\right)$. First we prove a theorem, stating that if ${\mathbf{G}}_{\mathbf{P}}$ is not positive definite on the action vector fields for the tangential action along ${\mu_G}^{-1}(0)$ then $\left({\mathrm{B}},g_{\mathrm{B}}\right)$ acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize self-parallel vector fields on ${\mathrm{M}}$ in terms of the positivity of the ${\mathbf{G}}_{\mathbf{P}}$-length of their tangential lifts along certain canonical subsets of $T{\mathrm{M}}$. We use this to derive some necessary conditions, on $G$ and ${\mathrm{M}}$, for actions to be tangentially positive on relevant subsets of $T{\mathrm{M}}$, which we then apply to isometric actions on complete conjugate point-free reducible Riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions.