Universität Stuttgart
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Item Open Access Stability of compact symmetric spaces(2022) Semmelmann, Uwe; Weingart, GregorIn this article, we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.Item Open Access Deformations of nearly G2 structures(2021) Nagy, Paul‐Andi; Semmelmann, UweWe describe the second order obstruction to deformation for nearly G2 structures on compact manifolds. Building on work of Alexandrov and Semmelmann, this allows proving rigidity under deformation for the proper nearly G2 structure on the Aloff–Wallach space N(1,1).Item Open Access Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds(2022) Schwahn, PaulAny 6-dimensional strict nearly Kähler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein-Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on F1,2=SU(3)/T2are not integrable into a curve of Einstein metrics.Item Open Access Sixteen-dimensional locally compact planes of Lenz-type V on which SU2H acts as a group of collineations(2020) Hähl, Hermann; Meyer, EkkehardWe explicitly construct all the planes mentioned in the title.Item Open Access Lines on K3 quartic surfaces in characteristic 3(2022) Veniani, Davide CesareWe investigate the number of straight lines contained in a K3 quartic surface X defined over an algebraically closed field of characteristic 3. We prove that if X contains 112 lines, then X is projectively equivalent to the Fermat quartic surface; otherwise, X contains at most 67 lines. We improve this bound to 58 if X contains a star (ie four distinct lines intersecting at a smooth point of X). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.Item Open Access Stability of Einstein metrics on symmetric spaces of compact type(2021) Schwahn, PaulWe prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces SU(n), n≥3, and E6/F4. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.Item Open Access Algebraic conditions for conformal superintegrability in arbitrary dimension(2024) Kress, Jonathan; Schöbel, Konrad; Vollmer, AndreasWe consider second order (maximally) conformally superintegrable systems and explain how the definition of such a system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant interpretation of superintegrability. Conformal equivalence in this context is a natural extension of the classical (linear) Stäckel transform, originating from the Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.Item Open Access On the ergodicity of the frame flow on even-dimensional manifolds(2024) Cekić, Mihajlo; Lefeuvre, Thibault; Moroianu, Andrei; Semmelmann, UweIt is known that the frame flow on a closed n-dimensional Riemannian manifold with negative sectional curvature is ergodic if nis odd and n≠7. In this paper we study its ergodicity in the remaining cases. For neven and n≠8,134, we show that: if n≡2mod 4 or n=4, the frame flow is ergodic if the manifold is ∼0.3-pinched, if n≡0mod 4, it is ergodic if the manifold is ∼0.6-pinched. In the three dimensions n=7,8,134, the respective pinching bounds that we need in order to prove ergodicity are 0.4962..., 0.6212..., and 0.5788.... This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that 0.25-pinched even-dimensional manifolds have an ergodic frame flow.Item Open Access Totally geodesic submanifolds and polar actions on Stiefel manifolds(2024) Gorodski, Claudio; Kollross, Andreas; Rodríguez-Vázquez, AlbertoWe classify totally geodesic submanifolds of the real Stiefel manifolds of orthogonal two-frames. We also classify polar actions on these Stiefel manifolds, specifically, we prove that the orbits of polar actions are lifts of polar actions on the corresponding Grassmannian. In the case of cohomogeneity one actions we are able to obtain a classification for all real, complex and quaternionic Stiefel manifolds of k-frames.