On Riemannian Foliations over Positively Curved Manifolds

An intrinsic curvature condition for submersions over Riemannian manifolds

“…The main tool used here is dual holonomy fields (as introduced in [Spe16]). Let c be a horizontal curve, i.e., ċ ∈ H, recall that a holonomy field ξ along c is a vertical field satisfying ∇ ċξ = −A * ċ ξ − S ċξ, where S : H × H → V is the second fundamental form of the fibers:…”

“…As a last observation, we recall an identity in [Spe16]. Given a Riemannian metric g, let K g (X, ν) = R g (X, ν, ν, X) be the unreduced sectional curvature of X ∧ ν.…”

“…On one hand, Lemma 2.2 guarantees that A * ċ ν(t) = 0 for the dual holonomy field ν along c with ν(0) = ν 0 . On the other hand, holonomy fields have bounded norm whenever π admits a compact structure group, i.e., there is a constant L such that ν(t) 2 ≤ L ν(0) 2 for every t (see [Spe16,Proposition 3.4]). Applying equation (2.4), we get d 2 dt 2 ν(t) 2 ≥ 2K g ( ċ, ν) ≥ κ ν(t) 2 for some fixed constant κ > 0, contradicting the boundedness of ν(t) 2 .…”

The curvature of convex sum of metrics and applications

“…Therefore, in order to possibly construct a metric of positive sectional curvature in a product manifold, we could, for instance, take a warped product (a particular type of general vertical warping). It is worth pointing out that, even though we were able to increase the curvature at every point in the torus, an arbitrary general vertical warping may produce planes with negative sectional curvature (see [Spe16], for example).…”

A note on the Petersen-Wilhelm conjecture

“…[17] and [5, p. 167]). We refer the reader to [14,20] for recent progress towards the conjecture in the general case, and to [12] for a complete list of all the (few) currently known Riemannian submersions between positively curved manifolds.…”