Spectrally distinguishing symmetric spaces II

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are ‘larger’ groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple. Using ‘Helgason spheres’, S(α), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G p(K n ), K=ℂ, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L′ be the geometric transformation group of M. Let L={ϕ: M→M: ϕ is a diffeomorphism and ϕ preserves arithmetic distance}. Then L=L′

We determine the stabilityof totally geodesic submanifolds in a compact symmetric space, which are called polars and meridians (see 2.1). These subspaces were introduced by Chen and Nagano ([CN-1]) and we have many interesting results after that ([CN-2], [N-l], [N-2], [NS-1], [NS-2] and [NS-3]). Recently, several results have been obtained about the stabilityof totally geodesic submanifolds in compact symmetric spaces. Ohnita gave the formula for the index, the nullityand the Killing nullity of a totally geodesic submanifold in a compact symmetric space in [0], in which he also proved that the Helgason sphere in a compact symmetric space is stable. Tasaki proved that the Helgason sphere in a compact Lie group is homologically volume-minimizing in its real homology classin [Ts-1]. He used the calibrationtheory. And there are studies about the stability of certain closed subgroups in a compact Lie group by Mashimo and Tasaki ([MT-1] and [MT-2]). Mashimo determined all the unstable Cartan embeddings of compact symmetric spaces in [M]. And there is a result about the stabilityof symmetric i?-spaces in Hermitian symmetric spaces and totally complex submanifolds in quaternionic Kahler symmetric spaces of classical type by Takeuchi ([Tk-2]). Recently Nagano and the author have obtained a result on a relationshipbetween the stabilityof minimal submanifolds and that of /^-harmonic maps ([NS-3]). In the present paper we study the stability of all the polars and meridians in every compact symmetric space by using Ohnita's method in Section 3. We will also study the stabilityof totally complex totallygeodesic submanifolds in quaternionic Kahler symmetric spaces of exceptional type in Section 4. I should like to express my gratitude to Professor T. Nagano for his useful advice and kindly support. This is the author's doctoral dissertation submitted to Sophia University in October, 1992.

A unit speed curve γ = γ(s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along γ and a positive constant k such that ∇sγ′(s) = kY(s), ∇sY(s) = −kγ′(s). A maximal totally geodesic sphere with maximal sectional curvature in a compact irreducible symmetric space M is called a Helgason sphere. A circle which lies in a Helgason sphere of a compact symmetric space is called a Helgason circle. In this article we establish some fundamental relationships between Helgason circles, Helgason spheres of irreducible symmetric spaces of compact type and the theory of immersions of finite type.

Let h h be a harmonic function defined on a spherical disk. It is shown that Δ k | h | 2 \Delta ^k |h|^2 is nonnegative for all k ∈ N k\in \mathbb {N} where Δ \Delta is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined on a disk in a normal homogeneous compact Riemannian manifold, and in particular in a symmetric space of the compact type. This complements a similar property for harmonic functions on R n \mathbb {R}^n discovered by the first two authors and is related to strong convexity of the L 2 L^2 -growth function of harmonic functions.

The volume of geodesic balls and tubes about totally geodesic submanifolds in compact symmetric spaces

In this paper, we study homogeneous Einstein-like metrics on the compact irreducible symmetric space M, which is not isometric to a compact Lie group and has rank greater than 1. Whenever there exists a closed proper subgroup G′ of G = Isom0(M) acting transitively on M, we find all G′-invariant $${\mathcal A}$$ -metrics and $${\mathcal B}$$ -metrics on M. More precisely, we prove that G′-invariant metrics on M must be $${\mathcal A}$$ -metrics, and G′-invariant $${\mathcal B}$$ -metrics on M are always Einstein.

Suppose \({\mathbb{A}}\) and \({\mathbb{B}}\) are normed division algebras, i.e. \({\mathbb{R}, \mathbb{C}, \mathbb{H}}\) or \({\mathbb{O}}\), we introduce and study Grassmannians of linear subspaces in \({(\mathbb{A}\otimes\mathbb{B})^{n}}\) which are complex/Lagrangian/maximal isotropic with respect to natural two tensors on \({(\mathbb{A}\otimes\mathbb{B})^{n}}\). We show that every irreducible compact symmetric space must be one of these Grassmannian spaces, possibly up to a finite cover. This gives a simple and uniform description of all compact symmetric spaces. This generalizes the Tits magic square description for simple Lie algebras to compact symmetric spaces.

This paper is a survey of our recent works on biharmonic homogeneous submanifolds in compact symmetric spaces (Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups (in preparation), Biharmonic homogeneous hypersurfaces in compact symmetric spaces. Differ Geom Appl 43, 155–179 (2015)) [12, 13]. We give a necessary and sufficient condition for an isometric immersion whose tension field is parallel to be biharmonic. By this criterion, we study biharmonic orbits of commutative Hermann actions in compact symmetric spaces, and give some classifications.

Minimal symmetric submani folds in compact Riemannian symmetric spaces

We assemble the basic facts required to discuss c~-functions of compact symmetric spaces from the representation-theoretic viewpoint, in principle, everything here in w 1 is contained in Garth Warner's book [6], and we refer to Warner [6] and Helgason [4] for the original sources (of which Caftan [3] is the principal one). Fix a compact riemannian symmetric space M and let G be the largest connected group of isometries. Thus G is a compact connected Lie group with an involutive automorphism a, and M = G/K where K is an open subgroup of G~= {geG :a(g) =g}, and the riemannian metric on M derives from a positive definite invariant bilinear form on the Lie algebra of G. (~ denotes the set of all equivalence classes In] of irreducible unitary representations n of G. Given In], V~ denotes the (finite dimensional complex Hilbert) space on which n represents G. A class [n]e(~ is of class 1 relative to K if there exists

The quotient of a Hermitian symmetric space of non-compact type by a torsion-free cocompact arithmetic subgroup of the identity component of the group of isometries of the symmetric space is called an arithmetic fake compact Hermitian symmetric space if it has the same Betti numbers as the compact dual of the Hermitian symmetric space. This is a natural generalization of the notion of “fake projective planes” to higher dimensions. Study of arithmetic fake compact Hermitian symmetric spaces of type An with even n has been completed in [PY1], [PY2]. The results of this paper, combined with those of [PY2], imply that there does not exist any arithmetic fake compact Hermitian symmetric space of type other than An, n ≤ 4 (see Theorems 1 and 2 in the Introduction below and Theorem 2 of [PY2]). The proof involves the volume formula given in [P], the Bruhat-Tits theory of reductive p-adic groups, and delicate estimates of various number theoretic invariants.

On gap rigidity problems for compact Hermitian symmetric spaces

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Frechet manifold of all embeddings of \(M\) in \(N\), endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\). The Frechet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\), inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\)). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\). We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.

On a theorem of Chernoff on rank one Riemannian symmetric spaces

Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space . It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.