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We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral description, i.e., can be described by an equivariant linear matrix inequality. Second, we show that the bijection induced by Kostant's Convexity Theorem between convex subsets invariant under a polar representation and convex subsets of a section invariant under the Weyl group preserves the classes of convex semi-algebraic sets, spectrahedral shadows, and rigidly convex sets., Comment: LaTeX2e, 17 pages

We prove a version of Weyl's Law for the basic spectrum of a closed singular Riemannian foliation $(M,\mathcal{F})$ with basic mean curvature. In the special case of $M=\mathbb{S}^n$, this gives an explicit formula for the volume of the leaf space $\mathbb{S}^n/\mathcal{F}$ in terms of the algebra of basic polynomials. In particular, $\operatorname{Vol}(\mathbb{S}^n/\mathcal{F})$ is a rational multiple of $\operatorname{Vol}(\mathbb{S}^m)$, where $m=\dim (\mathbb{S}^n/\mathcal{F})$., Comment: 27 pages, 1 figure

Given a compact Lie group $G$ and an orthogonal $G$-representation $V$, we give a purely metric criterion for a closed subset of the orbit space $V/G$ to have convex pre-image in $V$. In fact, this also holds with the natural quotient map $V\to V/G$ replaced with an arbitrary submetry $V\to X$. In this context, we introduce a notion of "fat section" which generalizes polar representations, representations of non-trivial copolarity, and isoparametric foliations. We show that Kostant's Convexity Theorem partially generalizes from polar representations to submetries with a fat section, and give examples illustrating that it does not fully generalize to this situation., Comment: 12 pages

We prove several new restrictions on the Euler characteristic and signature of oriented 4-manifolds with (positively or negatively) pinched sectional curvature. In particular, we show that simply connected 4-manifolds with $\delta\leq \sec\leq 1$, where $\delta=\frac{1}{1+3\sqrt3}\cong 0.161$, are homeomorphic to $S^4$ or $\mathbb C P^2$., Comment: LaTeX2e, 37 pages, 2 figures

We show Laplacian algebras are maximal, and give applications to the Classical Invariant Theory of real orthogonal representations of compact groups, including: The solution of the Inverse Invariant Theory problem for finite groups. An if-and-only-if criterion for when a separating set is a generating set. And the introduction of a class of generalized polarizations which, in a certain class of representations (including all representations of finite groups), always generates the algebra of invariants of their diagonal representations., Comment: 19 pages; Title, abstract, and order of results in the Introduction changed in order to emphasize the main result (maximality of Laplacian algebras); other small changes in exposition in various places; no changes in the actual results. Version accepted for publication by AMPA

Given an orthogonal representation of a compact group, we show that any element of the connected component of the isometry group of the orbit space lifts to an equivariant isometry of the original Euclidean space. Corollaries include a simple formula for this connected component, which applies to ``most'' representations., Comment: 9 pages. Final version, to appear in the Bulletin of the London Mathematical Society. Compared to the previous version, one of the corollaries to the main result has been reformulated as an easy-to-use formula (Proposition 5(b)). Moreover, many improvements to the presentation have been made, including the addition of a new section (Section 2)

Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence between such manifold submetries and maximal Laplacian algebras, thus solving the Inverse Invariant Theory problem for this class of partitions. Moreover, a solution to the analogous problem is provided for two smaller classes, namely orthogonal representations of finite groups, and transnormal systems with closed leaves., Comment: 33 pages. Final version. To appear in GAFA

We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $n\geq5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $n\geq5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra., Comment: LaTeX2e, 28 pages, final (revised) version. To appear in SIAM J. Appl. Algebra Geom

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant., Comment: 24 pages. The presentation has been improved following suggestions by an anonymous referee. Version accepted for publication by the Journal of the European Mathematical Society (JEMS)

We find many examples of compact Riemannian manifolds $(M,g)$ whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric $g$ is replaced with $g'$ in a neighbourhood of $g$. Our examples $(M,g)$ consist of certain minimal isoparametric hypersurfaces of spheres; their focal manifolds; the Lie groups $SU(n)$ for $n\leq 17$, and $Sp(n)$ for all $n$; and all quaternionic Grassmannians., Comment: 22 pages

We define virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second fundamental form, which is in general not symmetric. We prove that a manifold admits a virtual immersion with skew symmetric second fundamental form, if and only if it is a symmetric space, and in this case the virtual immersion is essentially unique., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1708.05881

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also give an application of the Bochner technique to closed $4$-manifolds with indefinite intersection form and $\sec>0$ or $\sec\geq0$, obtaining new insights into the Hopf Conjecture, without any symmetry assumptions., Comment: LaTeX2e, 25 pages, final version. To appear in Indiana Univ. Math. J

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSes, we improve this to a lower bound on the index alone, which is affine in the first Betti number. To prove these, we introduce a generalization of isometric immersions in Euclidean space. Compact symmetric spaces admit (and in fact are characterized by) such a structure with skew-symmetric second fundamental form., Comment: Version accepted for publication in Calc. Var. Partial Differential Equations

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite., Comment: LaTeX2e, 14 pages, revised version (some details given in previous versions are now omitted, following referee's request). To appear in Math. Ann

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds $W^6$, $W^{12}$ and $W^{24}$, which are respectively the manifolds of complete flags in $\mathbb C^3$, $\mathbb H^3$ and $\mathbb{Ca}^3$. Together with our earlier work, this concludes the classification of simply-connected homogeneous spaces that admit a homogeneous metric with strongly positive curvature., Comment: LaTeX2e, 15 pages, 1 figure, final version. To appear in Math. Z

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, and that most compact homogeneous spaces with positive sectional curvature satisfy it., Comment: LaTeX2e, 26 pages

Let $M$ be a Riemannian manifold with a polar action by the Lie group $G$, with section $\Sigma\subset M$ and generalized Weyl group $W$. We show that restriction to $\Sigma$ is a surjective map from the set of smooth $G$-invariant tensors on $M$ onto the set of smooth $W$-invariant tensors on $\Sigma$. Moreover, we show that every smooth $W$-invariant Riemannian metric on $\Sigma$ can be extended to a smooth $G$-invariant Riemannian metric on $M$ with respect to which the $G$-action remains polar with the same section $\Sigma$., Comment: arXiv admin note: text overlap with arXiv:1205.4765

We deal with a Lie group G acting by isometries on a Riemannian manifold M, such that the quotient M/G is an orbifold, or, equivalently, all slice representations are polar. We show that any smooth orbifold symmetric 2-tensor on M/G lifts to a smooth G-invariant symmetric 2-tensor on M. The proof relies on a fact about the Invariant Theory of finite reflection groups which may be of independent interest., Comment: 24 pages