Curvatures of Tangent Hyperquadric Bundles - Google Sites

Curvatures of Tangent Hyperquadric Bundles Takamichi Satoh (Tohoku University) [email protected] Joint work with Masami Sekizawa

The Sasaki metric can be determined on the tangent bundle T M over a pseudoRiemannian manifold (M, g). For r > 0 and  = ±1, tangent hyperquadric bundles of radius r are hypersurfaces of T M defined by { } T√r M = (x, u) ∈ T M | gx (u, u) = r2 , on which we induce a pseudo-Riemannian metric from the Sasaki metric. Kowalski-Sekizawa [1] have shown how the geometry of the tangent sphere bundle Tr M over a Riemannian manifold (M, g) depends on the radius r. We generalize a part of their results and obtain a new kind of results. First, we consider the scalar curvature of the tangent hyperquadric bundles. Theorem 1. Let (M, g), dim M ≥ 3, be a pseudo-Riemannian manifold with bounded sectional curvature. Then, for each sufficiently small positive number r, the tangent hyperquadric bundle T√r M is a space of positive scalar curvature if  = 1 and of negative scalar curvature if  = −1. In particular, these scalar curvatures are constant if (M, g) is non-Riemannian. Next, we consider the case that M has a constant sectional curvature. Theorem 2. Let (M, g), dim M ≥ 2, be a pseudo-Riemannian non-flat manifold of constant sectional curvature. Then, for each sufficiently large positive number r, the tangent hyperquadric bundle T√r M is a space of negative constant scalar curvature if  = 1 and of positive constant scalar curvature if  = −1. Combining above results, we obtain the following. Corollary 3. Let (M, g), dim M ≥ 3, be a pseudo-Riemannian non-flat manifold of constant sectional curvature. Then there exists a tangent hyperquadric bundle T√r M whose scalar curvature is preassigned constant.

References [1] O. Kowalski and M. Sekizawa, On tangent sphere bundles with small or large constant radius, Special issue in memory of Alfred Gray (1939–1998), Ann. Global Anal. Geom. 18 (2000), no. 3–4, 207–219.