An approach to understanding the geometry and topology of a given Riemannian manifold $M$ is to simplify the problem by ``reducing" $M$ via its ``symmetries''. This approach becomes quite relevant when $M$ has a ``large degree'' of symmetry. An appealing feature behind this approach is that we are free to choose the meaning behind the notion of symmetry, and degree. As a notion of symmetry we consider a particular family of foliations, which admit a ``bundle like metric'', which are known as *singular Riemannian foliations*. They are a natural candidate for symmetries of a manifold, since they encompass both the notion of *group actions by isometries* and *Riemannian submersion*s, as well as solutions to PDE which are compatible with the given Riemannian metric (for example the so called *isoperimetric foliations*). This fits in the context of the so called *Grove Symmetry Program*. In this program, Grove has proposed to first consider manifolds of positive curvature with a high degree of symmetry, i.e. with a large Lie group, acting by isometries.

In the setting of the Grove Symmetry Program, and more generally in the theory of compact transformation groups, a lot of attention has been devoted to torus actions. An extension for singular Riemannian foliations is to consider foliations where the leaves are aspherical, the so called *$A$-foliations*. In this case, all the homotopy information of a leaf is contained in its fundamental group. For the particular case when $M$ is simply-connected, the leaves of highest dimension are all homeomorphic to a torus, and so these foliations resemble torus actions.

Another context in which torus actions arise is that of ``collapse'' of Riemannian manifolds. That is a procedure where we deform the geometry of the manifold to obtain, at the end, a new, possibly not smooth, metric space of lower dimension. A central object in the study of collapse with bounded curvature is that of an $F$-structure. Moreover, this is an object that generalizes the notion of local torus actions on a smooth manifold.

Having as a starting point the theories of collapse with bounded curvature and singular Riemannian foliations, the present project studies *the compatibility of a singular Riemannian foliation and the notion of collapse*. In particular we focus on $A$-foliations on simply-connceted manifolds.

## Publications

We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space \(X\), or a singular Riemannian foliation on a manifold \(M\) is a closed subgroup of the isometry group of \(X\) in the case of a metric foliation, or of the isometry group of \(M\) for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.

**Related project(s):****43**Singular Riemannian foliations and collapse

We study \(\mathsf{RCD}\)-spaces \((X,d,\mathfrak{m})\) with group actions by isometries preserving the reference measure \(\mathfrak{m}\) and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that each slice at a point is homeomorphic to a non-negatively curved \(\mathsf{RCD}\)-space. Under the assumption that *\(X\)* is non-collapsed we further show that the slices are homeomorphic to metric cones over homogeneous spaces with \(\mathrm{Ric}\geq 0\). As a consequence we obtain complete topological structural results and a principal orbit representation theorem. Conversely, we show how to construct new \(\mathsf{RCD}\)-spaces from a cohomogeneity one group diagram, giving a complete description of \(\mathsf{RCD}\)-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed \(\mathsf{RCD}\)-spaces of essential dimension at most 4.

**Related project(s):****43**Singular Riemannian foliations and collapse

We present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension on a compact manifold, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that a closed flat regular Riemannian foliation with leaves of positive dimension on a compact simply-connected manifold is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions.

**Related project(s):****43**Singular Riemannian foliations and collapse

A singular foliation \(\mathcal{F}\) on a complete Riemannian manifold \(M\) is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of *M* into orbits of an isometric action. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRF with special properties, e.g. isoparametric foliations, SRF on fibers bundles with Sasaki metric, and orbit-like foliations. More precisely, we prove two results analogous to Palais' Principle of Symmetric Criticality, one is a general principle for \(\mathcal{F}\) symmetric operators on the Hilbert space \(W^{1,2}(M)\), the other one is for \(\mathcal{F}\) symmetric integral operators on the Banach spaces \(W^{1,p}(M)\). These results together with a \(\mathcal{F}\) version of Rellich Kondrachov Hebey Vaugon Embedding Theorem allow us to circumvent difficulties with Sobolev's critical exponents when considering applications of Calculus of Variations to find solutions to PDEs. To exemplify this we prove the existence of weak solutions to a class of variational problems which includes \(p\)-Kirschoff problems.

**Related project(s):****43**Singular Riemannian foliations and collapse

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian (*????*+2)-manifold, with regular leaves homeomorphic to the *n*-torus, is given by a smooth effective *n*-torus action. This solves in the negative for the codimension 2 case a question about the existence of foliations by exotic tori on simply-connected manifolds.

Journal | Mathematische Zeitschrift |

Publisher | Springer |

Volume | 304 |

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**Related project(s):****43**Singular Riemannian foliations and collapse

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

Journal | Calculus of Variations and Partial Differential Equations |

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**Related project(s):****43**Singular Riemannian foliations and collapse

We expand upon the notion of a pre-section for a singular Riemannian foliation \((M,\mathcal{F})\), i.e. a proper submanifold \(N\subset M\) retaining all the transverse geometry of the foliation. This generalization of a polar foliation provides a similar reduction, allowing one to recognize certain geometric or topological properties of \((M,\mathcal{F})\) and the leaf space \(M/\mathcal{F}\). In particular, we show that if a foliated manifold \(M\) has positive sectional curvature and contains a non-trivial pre-section, then the leaf space \(M/\mathcal{F}\) has nonempty boundary. We recover as corollaries the known result for the special case of polar foliations as well as the well-known analogue for isometric group actions.

Journal | Ann. Global Anal. Geom. |

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**Related project(s):****43**Singular Riemannian foliations and collapse

Journal | Journal of Geometric Analysis |

Volume | 32 |

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Link to published version |

**Related project(s):****43**Singular Riemannian foliations and collapse

## Team Members

**Dr. Diego Corro**

Project leader

Cardiff University

corrotapiad(at)cardiff.ac.uk