Some Formulas and Tables

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.

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.

This paper is a survey of results (partly obtained by the authors) on homogeneous spaces of Lie groups \(G\) with a compact stabilizer subgroup \(H\), on which every \(G\)-invariant distribution is integrable. It is proved that the condition of integrability is necessary and sufficient for every invariant inner metric to be (holonomic) Finsler on such a space. As a corollary of the obtained results, we assert that the class of homogeneous spaces with invariant non-holonomic Riemannian metrics (in other terms, sub-Riemannian or Carnot–Caratheodory metrics), which were actively studied last 3 decades after Gromov’s work, is rather broad. On the other hand, the class of homogeneous spaces with integrable invariant distributions includes Cartan’s symmetric spaces as well as isotropy irreducible, in particular, strictly isotropy irreducible, homogeneous spaces, which have been classified in simply connected case in the papers by Wang and Ziller (respectively, by Manturov, Wolf and Kramer). Special attention is paid to the case, when the Lie groups \(G\) and \(H\) are connected. Then the integrability condition of the invariant distributions is equivalent to a purely algebraic condition, that for the Lie algebra \(h\) of the subgroup \(H\), any \(ad(h)\)-invariant vector subspace in the Lie algebra \(g\) of the Lie group \(G\) is a Lie subalgebra; such Lie subalgebra \(h\subset g\) is called a strong subalgebra. The first author proved that a simply connected and compact space \(G/H\) with this condition is isomorphic to a direct product of strictly isotropy irreducible homogeneous spaces. In line with this, the second author recently found several non-compact simply connected homogeneous spaces with this condition, which are not isomorphic to such direct products. These results are naturally related to the structure questions of a class of general homogeneous locally compact spaces with an inner metric. This class is exactly the closure in the Gromov–Hausdorff sense of the class of homogeneous manifolds with an inner metric. Any such manifold is isometric to some homogeneous manifold \(G/H\) with \(G\)-invariant (may be, non-holonomic) Finsler metric. The authors give fairly detailed survey of the existing methods of the search of geodesics, i.e., locally shortest arcs, on such manifolds (particularly, with invariant non-holonomic Riemannian metrics), non-holonomic metric geometry and its relations with the geometric group theory, \(CR\)-manifolds, thermodynamics, etc. Some unsolved problems are suggested.

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

Abstract. 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 [1] we proved that i(M) is bounded from below by the rank rk(M) of M, that is, rk(M) ≤ i(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 R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with i(M) ∈ {4, 5, 6}. 1.

The metric is an important tensorial object that introduces more structure into a (differential) manifold. The metric coefficients allow for example to determine the length of parameter curves in the manifold and make it possible to relate corresponding co- and contravariant objects defined on the manifold. Thus the inner product of either co- or contravariant quantities and in particular the angle between two vectors or (covectors) may be computed. Adopting the Ricci postulate of vanishing covariant derivative of the metric coefficients results in a decomposition of the fully covariant connection into its Riemann part, that depends exclusively on the metric, and the contortion, that depends exclusively on the torsion. Moreover, in this case the length of vectors and their angle with respect to geodesics are preserved upon parallel transport. Based on the metric coefficients the fully covariant curvature tensor, displaying left and right skew symmetry, is introduced and the corresponding modifications of the Bianchi identities are highlighted. Likewise the metric allows to introduce the mixed-variant Ricci tensor together with corresponding identities. Finally the metric enables to compute the Ricci scalar from the previous curvature tensors. The particular case of a symmetric, metrically connected manifold represents a Riemann geometry with associated Riemann curvature tensor as fundamental for example in the Einstein Theory of General Relativity/Gravitation. A generalization is obtained by allowing for metric manifolds with non-vanishing covariant derivative of its metric coefficients. The presence of the non-metricity is then subsequently reflected by extra terms in the decomposition of the connection and in the explicit representation of the curvature tensor, the Ricci tensor, the Ricci scalar, and the corresponding Bianchi identities for the various curvature quantities.

Let V be a finite-dimensional real vector space and K a compact simple Lie group with Lie algebra \mathfrak{k} . Consider the Fréchet–Lie group G := J_0^\infty(V; K) of \infty -jets at 0\in V of smooth maps V \to K , with Lie algebra \mathfrak{g}=J_0^\infty(V; \mathfrak{k}) . Let P be a Lie group and write \mathfrak{p}:=\operatorname{Lie}(P) . Let \alpha be a smooth P -action on G . We study smooth projective unitary representations \bar{\rho} of G\rtimes_\alpha P that satisfy a so-called generalized positive energy condition. In particular, this class captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by \bar{\rho}(G) . We show that this condition imposes severe restrictions on the derived representation d\bar{\rho} of \mathfrak{g}\rtimes \mathfrak{p} , leading in particular to sufficient conditions for {\bar{\rho}}|_{G} to factor through J_0^2(V; K) , or even through K .

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kahler metric onG/T does not have volume-preserving geodesic symmetries. From the Kahler/de Rham decomposition of a compact homogeneous Kahler manifold [8], our result extends to the invariant Kahler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kahler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω‎ on M, by averaging all the left translates of ω‎ over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.

Gradings on classical central simple real Lie algebras

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

This book provides the first unified examination of the relationship between Radon transforms on symmetric spaces of compact type and the infinitesimal versions of two fundamental rigidity problems in Riemannian geometry. Its primary focus is the spectral rigidity problem: Can the metric of a given Riemannian symmetric space of compact type be characterized by means of the spectrum of its Laplacian? It also addresses a question rooted in the Blaschke problem: Is a Riemannian metric on a projective space whose geodesics are all closed and of the same length isometric to the canonical metric? The authors comprehensively treat the results concerning Radon transforms and the infinitesimal versions of these two problems. Their main result implies that most Grassmannians are spectrally rigid to the first order. This is particularly important, for there are still few isospectrality results for positively curved spaces and these are the first such results for symmetric spaces of compact type of rank >1. The authors exploit the theory of overdetermined partial differential equations and harmonic analysis on symmetric spaces to provide criteria for infinitesimal rigidity that apply to a large class of spaces. A substantial amount of basic material about Riemannian geometry, symmetric spaces, and Radon transforms is included in a clear and elegant presentation that will be useful to researchers and advanced students in differential geometry.

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form G/H, where G is a connected compact simple Lie group with an automorphism γ˜ of order four on G and H is a fixed points subgroup Gγ of G. According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian 4-symmetric spaces G/H in the case where G is of type E8. In the present article, we give the explicit form of automorphisms ω˜4, κ˜4 and ε˜4 of order four on E8 induced by the C-linear transformations ω4, κ4 and ε4 of the 248-dimensional C-vector space e8C, respectively. Further, we determine the structure of these fixed points subgroups (E8)ω4, (E8)κ4 and (E8)ε4 of E8. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces G/H above corresponding to the Lie algebras h=su(2)⊕iR⊕e6, iR⊕so(14) and h=su(2)⊕iR⊕so(12), where h=Lie(H). With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in E8.

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jimenez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of order four on $G$ and $H$ is a fixed points subgroup $G^\gamma$ of $G$. According to the classification by J. A. Jimenez, there exist seven compact simply connected Riemannian 4-symmetric spaces $G/H$ in the case where $G$ is of type $E_8$. In the present article, we give the explicit form of automorphisms $\tilde{w}_4$, $\tilde{\upsilon}_4$ and $\tilde{\mu}_4$ of order four on $E_8$ induced by the $C$-linear transformations $w_4, \upsilon_4$ and $\mu_4$ of the 248-dimensional vector space $𝔢^C_8$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8)^{w_4}, (E_8)^{\upsilon_4}$ and $(E_8)^{\mu_4}$ of $E_8$. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $G/H$ above corresponding to the Lie algebras $𝔥 = i \boldsymbol{R} \oplus 𝔰𝔲(8)$, $i\boldsymbol{R} \oplus 𝔢_7$ and $𝔥 = 𝔰𝔲(2) \oplus 𝔰𝔲(8)$, where $𝔥 = \mathrm{Lie}(H)$.

In this paper, we consider a Lie group $G$ equipped with two left-invariant Riemannian metrics $g^1$ and $g^2$. Using these two left-invariant Riemannian metrics we define a left-invariant Riemannian metric $\tilde{g}$ on the tangent Lie group $TG$. The Levi-Civita connection, tensor curvature, and sectional curvature of $(TG,\tilde{g})$ in terms of $g^1$ and $g^2$ are given. Also, we give a sufficient condition for $\tilde{g}$ to be bi-invariant. Finally, motivated by the recent work of D. N. Pham, using symplectic forms $\omega _1$ and $\omega _2$ on $G$ we define a symplectic form $\tilde{\omega}$ on $TG$.