Need Not Preserve Positive

If π : M → B is a Riemannian Submersion and M has positive sectional curvature, O’Neill’s Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature. In [2], O’Neill shows that if π : M → B is a Riemannian submersion, then secB (x, y) = secM (x̃, ỹ) + 3 |Ax̃ỹ| , where {x, y} is an orthonormal basis for a tangent plane to B, {x̃, ỹ} a horizontal lift of {x, y} and A is O’Neill’s A-tensor. An immediate observation is that if the sectional curvature of all two-planes in M are bounded below by k, then the same must be true for all two-planes in B. In particular, Riemannian submersions preserve positive sectional curvature. David Wraith asked us for an example of a Riemannian submersion that does not preserve positive Ricci curvature. While many people think that such an example must exist, we do not know of one in the literature but provide some here. Theorem 1. For any C > 0, there is a Riemannian submersion π : M → B for which M is compact with positive Ricci curvature and B has some Ricci curvatures less than −C. The examples are constructed as warped products S ×ν F , where F is any manifold that admits a metric with Ricci curvature ≥ 1, and the metrics on S are C–close to the constant curvature 1 metric. In his thesis, the first author provides examples where the base metric on S can be C–close to any predetermined, positively-curved, rotationally-symmetric metric on S, and F is any manifold that admits a metric with Ricci curvature ≥ 1. [7] Since the metrics on our base spaces are C–close to the constant curvature 1 metric on S and yet have a curvature that is≤ −C, the region of negative curvature is necessarily small, and in fact has measure converging to 0 as −C → −∞. In this sense, our examples are local. On the other hand, we show that a truly global example is not possible. Theorem 2. If M is a compact Riemannian manifold with positive Ricci curvature, then there is no Riemannian submersion π : M −→ B to a space B with nonpositive Ricci curvature. We prove Theorem 1 in Sections 1 and 2 and Theorem 2 in Section 3. We are grateful to Bun Wong for asking a question that lead us to Theorem 2, to Pedro Solórzano for assisting us with a calculation, and especially to David Wraith 1 2 CURTIS PRO AND FREDERICK WILHELM for insightful criticisms of the manuscript and asking us to provide the examples of Theorem 1. 1. Vertical Warping Given a Riemannian submersion π : M → B, the vertical and horizontal distributions are defined as V := ker π∗ and H := (ker π∗), respectively. This gives a splitting of the tangent bundle as TM = V ⊕H. If g is the metric on M , we denote by g and g the restrictions of g to H and V . Define a new metric gν := e g + g on M , where ν is any smooth function on B. Note that both H and g are unchanged, so π : (M, gν) → B is also Riemannian. The calculations that give important geometric quantities associated to gν in terms of g and ν are carried out in [1] on page 45. In particular, the (0, 2) Ricci tensor Ricν of gν is given in detail. When M = B m×F k and g is a product metric, these quantities reduce to the following (Corollary 2.2.2 [1]). For horizontal X,Y and vertical U, V , we have Ricν(X,Y ) = RicB(X,Y )− k(Hess ν(X,Y ) + g(∇ν,X)g(∇ν, Y )), (1.1) Ricν(X,U) = 0, and (1.2) Ricν(U, V ) = RicF (U, V )− g(U, V )e(∆ν + k|∇ν|). (1.3) (There is a sign error in the analog of Equation 1.3 in [1].) We denote fields on B and their horizontal lifts via π1 : B ×F → B by the same letter. We write B ×ν F to denote the warped product metric gν on B × F . 2. The Warped Product S φ ×ν F Choose φ : [0, π] → [0,∞) so that S with the metric gφ = dr + φdθ is a smooth Riemannian manifold denoted by S φ. Let ν : [0, π] → R be a function on S φ that only depends on r. Consider the warped product S 2 φ ×ν F where (F, gF ) is any k-dimensional manifold (k ≥ 2) with RicF ≥ 1. Using the notation ν̇ = ∂rν, since ν only depends on r, ∇ν = ν̇∂r. If L is the Lie derivative, we have, 2Hess ν = L∇νgφ = Lν̇∂rgφ = ν̇L∂rgφ + dν̇dr + drdν̇ = 2ν̇Hess r + 2ν̈dr. Thus the Hessian of ν is given by Hess ν = ν̈dr + ν̇φφ̇dθ. The Ricci tensor of S φ is given as RicS2 φ = − φ̈ φ gφ. (see [4], p.69) RIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE 3 Let Richν and Ric v ν denote Ricν restricted to the horizontal and vertical distribution, respectively. Equation (1.1) can be written as (2.1) − Richν = [