Tangentially positive isometric actions and conjugate points

Abstract:

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.