Biharmonic Homogeneous Submanifolds in Compact Symmetric Spaces

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.

We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducible symmetric spaces of compact type which are singular orbits of commutative Hermann actions of cohomogeneity two. Also, in compact simple Lie groups, we determine all the biharmonic hypersurfaces which are regular orbits of actions of the direct product of two symmetric subgroups which are associated to commutative Hermann actions of cohomogeneity one.

In thispaper we shall study minimal submanifolds in compact symmetric spaces and homologically volume minimizing submanifolds in compact simple Lie groups and quanternionic Kahler manifolds. The firstsubjectis studied by computing the second fundamental forms of submanifolds. In Section 2 using the structure theorem of the firstconjugate loci of compact symmetric spaces (Takeuchi [5])we compute the second fundamental form of a certainsubmanifold which is open and dense in the firstconjugate locus of a compact symmetric space and prove the minimality of it. Moreover we show that the submanifold has no geodesic point. The second subjectis studied by using the notion calibration introduced by Harvey and Lawson [2]. This notionisused in Sections 3 and 4. The fundamental 2-form of a Kahler manifold is one of important examples of calibrations. It satisfiesWirtinger's inequality, which can be statedas follows. Let M be a Kahler manifold with fundamental 2-form o). Then

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

Minimal symmetric submani folds in compact Riemannian symmetric spaces

In this paper, we give sufficient conditions for orbits of Hermann actions to be weakly reflective in terms of symmetric triads, that is a generalization of irreducible root systems. Using these sufficient conditions, we obtain new examples of weakly reflective submanifolds in compact symmetric spaces.

Stability of certain minimal submanifolds in compact symmetric spaces of rank two

In this note we consider two methods in order to investigate volume minimizing submanifolds in compact symmetric spaces. The first is calibration ([4]) and the second is integral geometry. We can show that certain submanifolds are volume minimizing in their real homology classes using calibrations. A calibration is a closed differential form on a Riemannian manifold which satisfies a certain inequality. A definition of calibrations will be given in Section 1. On the othor hand we can prove that certain submanifolds are volume minimizing in its homotopy classes using integral geometry. We shall use a generalized Poincare's formula in Riemannian homogeneous spaces given by Howard

In [1], J. Berndt and H. Tamaru classified all the cohomogeneity one actions on Riemannian symmetric spaces of noncompact type with a totally geodesic singular orbit. Also they provided that there is a one-to-one correspondence between the totally geodesic singular orbits of cohomogeneity one actions on a Riemannian symmetric space of noncompact type and those on its dual simply connected compact Riemannian symmetric space. In this paper, we determine stability of the totally geodesic singular orbits in simply connected compact symmetric spaces which obtained by the duality stated above.

In this paper, we give sufficient conditions for orbits of Hermann actions to be weakly reflective in terms of symmetric triads, that is a generalization of irreducible root systems. Using these sufficient conditions, we obtain new examples of weakly reflective submanifolds in compact symmetric spaces.

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′

The main result is an LP mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semisimple Lie group. A central element in the proof is a Lie group-Lie algebra analog of the theorems in classical Fourier analysis that allow one to pass back and forth between multiplier operators for Fourier series in several variables and multiplier operators for the Fourier transform in Euclidean space. To obtain the LP mean convergence theorem, the theory of the Hilbert transform with weight function is needed. Introduction. A theorem of M. Riesz says that if f is in LP of the circle, 1 < p < oo, and if SNf(x)__Nakeikx is the Nth partial sum of the Fourier series of f, then SN f converges to f in the LP norm as N -* . Pollard [15] proved a similar result for Jacobi polynomials on the interval [1, 1]. If fis inLP([-1, 1]; (1 -x)'(l -x)dx) and if N SN f (X) =E do ', Oa Rk ' (X) k=O is the Nth partial sum of the Jacobi series of f, then SN f converges to f in the LP norm provided 4 ma+ ? 1< <4min o+l 3 +1 Xe+ '2,B + 3, 2 F1 + 1 Here R'' 0 is a normalized Jacobi polynomial and dk' p is an appropriate constant. It is well known that for suitable choices of a, 3 the {R 'kn} are the elementary spherical functions for the rank 1 symmetric spaces of compact type. Cast in this setting Pollard's theorem is an LP mean convergence result for bi-K invariant functions on the rank 1 compact symmetric space U/K. In this paper we investigate extending this result to higher rank symmetric spaces. Even in the abelian case of the n-torus Tn the result depends drastically on how the multiple series is summed. Consider for example f in LP(Tn) with Received by the editors October 3, 1974 and, in revised form, March 12, 1975. AMS (MOS) subject classifications (1970). Primary 43A90, 43A75; Secondary 33A45, 33A75. 61 Copyright @ 1976, American Mathematical Society This content downloaded from 207.46.13.58 on Sat, 17 Sep 2016 05:41:43 UTC All use subject to http://about.jstor.org/terms

We construct a gauge theoretic change of variables for the wave map from R × R into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation n ≥ 4 for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4. 0. Introduction The wave map equation between two Riemannian manifoldsthe wave equation version of the evolution equations which are derived from the same geometric considerations as the harmonic map equation between two Riemannian manifoldshas been studied by a number of mathematicians in the last decade. The work of Klainerman and Machedon and Klainerman and Selberg [5] [6] [8] studying the Cauchy problem for regular data is probably the best known. The more recent work of Tataru [15], [16] and Tao [13] [14] relies and further develops deep ideas from harmonic analysis in Tao’s case in conjunction with gauge theoretic geometric methods and thus seems very promising. Keel and Tao studied the one (spatial) dimensional case in [4]. In [13], Tao established the global regularity for wave maps from R × R into the sphere S when n ≥ 5. Similar results to those of Tao were obtained by Klainerman and Rodniansky [7] for target manifolds that admit a bounded parallelizable structure. In this paper we are interested in revisiting this work. We study the Cauchy problem for wave maps from R × R into a (compact) Lie group (or Riemannian symmetric 1991 Mathematics Subject Classification. Primary 35J10, Secondary 45B15, 42B35.

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

ABSTRACTSome nonlinear stochastic differential games are formulated in the family of complex and quaternion projective spaces that are among the rank one compact symmetric spaces that consist of the spheres, the projective spaces over , and and one arising from an exceptional Lie algebra called the Cayley plane. The payoff functionals for the differential games are obtained from some eigenfunctions of the radial part of the Laplacians for these Riemannian manifolds and these payoffs induce symmetries for the game problems that reduce the required analysis to a radial direction in these manifolds. These projective spaces are given a natural Riemannian metric from the Killing forms of the compact Lie groups for these spaces that are the special unitary groups, SU(n), and the symplectic groups, Sp(n). Explicit optimal control strategies are obtained for these differential games and the explicit payoffs are given. A countable family of distinct solvable stochastic differential games can be obtained for each of these compact symmetric spaces.