SMOOTHNESS CONDITIONS IN COHOMOGENEITY ONE MANIFOLDS
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SMOOTHNESS CONDITIONS IN COHOMOGENEITY ONE MANIFOLDS L. VERDIANI∗
W. ZILLER∗∗
University of Firenze
University of Pennsylvania
[email protected]
[email protected]
Abstract. We present an efficient method for determining the conditions that a metric on a cohomogeneity one manifold, defined in terms of functions on the regular part, needs to satisfy in order to extend smoothly to the singular orbit.
Introduction A group action is called a cohomogeneity one action if its generic orbits are hypersurfaces. Such actions have been used frequently to construct examples of various types: Einstein metrics, soliton metrics, metrics with positive or nonnegative curvature and metrics with special holonomy. See [4], [6], [7], [8], [12] for a selection of such results. The advantage of such a metric is that geometric problems are reduced to studying its behavior along a fixed geodesic c(t) normal to all orbits. The metric is described by a finite collection of functions of t, which for each time specifies the homogeneous metric on the principal orbits. One aspect one needs to understand is what conditions these functions must satisfy if regular orbits collapse to a lower-dimensional singular orbit. These smoothness conditions are often crucial ingredients in obstructions, e.g., to non-negative or positive curvature, see [9], [14], [15]. The goal of this paper is to devise a simple procedure in order to derive such conditions explicitly. The local structure of a cohomogeneity one manifold near a collapsing orbit can be described in terms of Lie subgroups H ⊂ K ⊂ G with K/H = S` , ` > 0. The action of K on S` extends to a linear action on D = D`+1 ⊂ R`+1 and thus M = G ×K D is a homogeneous disc bundle, where K acts as (g, p) → (gk −1 , kp), and with boundary G×K ∂ D = G×K K/H = G/H a principal orbit. The Lie group G acts by cohomogeneity one on M by left multiplication in the first coordinate. A compact (simply connected) cohomogeneity one manifold is the union of two such homogeneous disc bundles. For simplicity we write M = G ×K V with V ' Rn . Given a smooth G invariant metric on the open dense set of regular points, i.e., the complement of the lower-dimensional singular orbit, the problem is when the extension of this metric to the singular orbit is smooth. We first simplify the DOI: 10.1007/S00031-020-09618-9 by PRIN and GNSAGA grants. Supported by a grant from the National Science Foundation. Received May 16, 2019. Accepted July 28, 2020. Corresponding Author: L. Verdiani, e-mail: [email protected] ∗ Supported ∗∗
L. VERDIANI, W. ZILLER
problem as follows: Theorem 1. Let G act by cohomogeneity one on M = G ×K V and g be a smooth cohomogeneity one metric defined on the set of regular points in M . Then g has a smooth extension to the singular orbit if and only if it is smooth when restricted to every 2-plane in the slice V containing c(0). ˙ As we will see, it follows from the classification of transitive actions on spheres, that it is sufficient to require the condition only f
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