In 1980, Onishchik introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank less than or equal to 2, but for higher rank it was unclear how to tackle the problem. In earlier papers we developed several approaches to this problem, which allowed us to calculate the index for many symmetric spaces. Our systematic approach led to a conjecture for how to calculate the index. The purpose of this paper is to verify the conjecture.
Let M be an irreducible Riemannian symmetric space. The index of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less or equal than 3.
We find the index of $\widetilde{\nabla}$-quasi-conformally symmetric and $\widetilde{\nabla}$-concircularly symmetric semi-Riemannian manifolds, where $\widetilde{\nabla}$ is metric connection.
The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold (Onishchik, 1980). There is a conjecture by the first two authors for how to calculate the index. In this paper we give an affirmative answer to this conjecture for the exceptional Riemannian symmetric spaces and for the classical symmetric spaces Sp(r,R)/U(r). Our methodology is new and based on the idea of using slice representations for studying totally geodesic submanifolds.
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric …
In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying some curvature conditions. By modifying Cheng-Yau's operator $\square$ given in {\cite{ChengYau77}}, we introduce a modified operator $L$ and give new estimates of $L(nH)$ and $\square(nH)$ of such spacelike hypersurfaces. Finally, we give partial generalizations of some conjectures in locally symmetric …
We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are g…
We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.
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.
Let $γ$ be a non-degenerate Ustilovsky geodesic in $Ham (M, ω)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $ γ$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, considered as periodic orbits.
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. The purpose of this note is to determine the index i(M) for all irreducible Riemannian symmetric spaces M of type (II) and (IV).
In this paper, we give a general group-theoretic construction of affine $\RR$-buildings, and more generally, of affine $Λ$-buildings, associated to semisimple Lie groups over nonarchimedean real closed fields. The construction of Kleiner-Leeb using the asymptotic cone of a Riemannian symmetric space appears as a special case. The explicit knowledge of the building arising here as the asymptotic cone simplifies the proof of the Margulis conjecture due to Kleiner-Leeb.
In this paper we study massless fermions coupled to spherically symmetric $SU(N)$ monopoles without Yukawa couplings between the Higgs and fermion fields. The corresponding Dirac operator is not Fredholm and the associated eigenfunctions are not $L^2$-normalizable. Here we derive a formula for the dimension of the plane-wave normalizable kernel of such a Dirac operator for fermions of any representation of $SU(N)$ in the presence of any spherically symmetric monopole background. Notably, our results also apply to fermions coupled to monopoles that preserve non-abelian gauge symmetry.
The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of th…
In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $λ$-Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This $λ$-Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal $λ$-Lipschitz rigidity theorem for positive scalar curvature metrics on hemispheres. These give positive answers to the corresponding open ques…
