Riemannian Submersions Need Not Preserve Positive Ricci Curvature

If $\pi :M\rightarrow B$ 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.

The examples are constructed as warped products S 2 × ν F , where F is any manifold that admits a metric with Ricci curvature ≥ 1, and the metrics on S 2 are C 1 -close to the constant curvature 1 metric. In his thesis, the first author provides examples where the base metric on S 2 can be C 1 -close to any predetermined, positively-curved, rotationally-symmetric metric on S 2 , and F is any manifold that admits a metric with Ricci curvature ≥ 1. [7] Since the metrics on our base spaces are C 1 -close to the constant curvature 1 metric on S 2 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 for insightful criticisms of the manuscript and asking us to provide the examples of Theorem 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 If g is the metric on M , we denote by g h and g v the restrictions of g to H and V. Define a new metric g ν := e 2ν g v + g h on M , where ν is any smooth function on B. Note that both H and g h are unchanged, so π : (M, g ν ) → B is also Riemannian.
The calculations that give important geometric quantities associated to g ν in terms of g and ν are carried out in [1] on page 45. In particular, the (0, 2) Ricci tensor Ric ν of g ν 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 There is a sign error in the analog of Equation 1.3 in [1].) We denote fields on B and their horizontal lifts via π 1 : B × F → B by the same letter. We write B × ν F to denote the warped product metric g ν on B × F .

The Warped Product
ϕ that only depends on r. Consider the warped product S 2 ϕ × ν F where (F, g F ) is any k-dimensional manifold (k ≥ 2) with Ric F ≥ 1. Using the notationν = ∂ r ν, since ν only depends on r, ∇ν =ν∂ r .
If L is the Lie derivative, we have, Thus the Hessian of ν is given by The Ricci tensor of S 2 ϕ is given as [4], p.69) Let Ric h ν and Ric v ν denote Ric ν restricted to the horizontal and vertical distribution, respectively. Equation (1.1) can be written as and equation (1.3) can be written as Notice that since Ric F ≥ 1, if Ric h ν is positive, then these equations together with Equation (1.2) imply that S 2 ϕ × ν+ln λ F has positive Ricci curvature, provided λ is a sufficiently small positive constant.
By requiring thatφ(p) > 0 for some point p ∈ (0, π), the projection π 1 : ϕ is a Riemannian submersion for which the base has points of negative Ricci curvature. Therefore, to describe a Riemannian submersion that does not preserve positive Ricci curvature, it suffices to find functions ϕ and ν so that (1) S 2 ϕ is smooth and has points of negative curvature, that is, (3) ν is constant in a neighborhood of 0 and π. For a C r -function f : R −→ R, we write The main idea is embodied in the following Lemma. Lemma 1. Given any sufficiently small η > 0, p ∈ (0, π/4), and ε > 0, there is a δ > 0 and smooth functions ϕ, ν : [0, π] → [0, ∞) with the following properties.
By composing with projection, π 1 : S 2 × F −→ S 2 , we can view r as a function on S 2 × F, which by abuse of notation we also call r.
On the other hand, along r −1 (p) , the Ricci curvature of S 2 ϕ is Sinceφ (p) = η and ϕ (p) is almost sin (p) , Remark 1. The proof above exploits the principle of Lemma 3.1 of [5] about which components of the curvature change a lot under a certain type of C 1 -small deformation. This principle will be applied in a revised version [6], where a revised version of Lemma 3.1 of [5] will appear.

The Base has a Positive Ricci Curvature
In this section, we prove Theorem 2 with an argument that is similar in spirit to the proof of the main theorem of [3]. We use the notation of [2] for the infinitesimal geometry of a submersion.
Suppose π : M −→ B is a Riemannian submersion with M compact, Ric M > 0 and Ric B ≤ 0. Exploiting the compactness of the unit tangent bundle T 1 M of M, we get Propositions 2 and 3, stated below. To pose them we need a metric space structure on T 1 M. Any metric that induces the topology of T 1 M will serve, the most geometrically relevant is perhaps the Sasaki metric (see [8]). Proposition 2. Let T 1 M, dist be the unit tangent bundle of M with any metric distance that induces the topology of T 1 M . For any ε > 0, there is a unit speed geodesic γ : [0, l] −→ M so that l ≥ 1 and dist (γ (0) ,γ (l)) < ε. Proposition 3. Let H 1 be the set of unit horizontal vectors for π. The map T : is uniformly continuous. Here T is O'Neill's T -tensor.
Proof of Theorem 2. Let γ be a horizontal, almost periodic geodesic as in Proposition 2. By re-scaling we may assume that Ric M ≥ 1. Then l ≤ l 0 Ric (γ,γ) .
be a vertically-parallel, orthonormal framing for the vertical distribution along γ, and let {E j } b j=1 be an orthonormal framing for the horizontal distribution along γ. Using O'Neill's vertizonal and horizontal curvature equations and the hypothesis that Ric B ≤ 0, we get l ≤ l 0 Ric (γ,γ) which gives a contradiction.