Locally compact homogeneous spaces with inner metric
aa r X i v : . [ m a t h . DG ] D ec LOCALLY COMPACT HOMOGENEOUS SPACES WITH INNERMETRIC
V. N. BERESTOVSKII
Abstract.
The author reviews his results on locally compact homoge-neous spaces with inner metric, in particular, homogeneous manifolds withinner metric. The latter are isometric to homogeneous (sub-)Finslerianmanifolds; under some additional conditions they are isometric to homo-geneous (sub)-Riemannian manifolds. The class Ω of all locally compacthomogeneous spaces with inner metric is supplied with some metric d BGH such that 1) (Ω , d BGH ) is a complete metric space; 2) a sequences in (Ω , d BGH ) is converging if and only if it is converging in Gromov-Hausdorffsense; 3) the subclasses M of homogeneous manifolds with inner metric and LG of connected Lie groups with left-invariant Finslerian metric are every-where dense in (Ω , d BGH ) . It is given a metric characterization of Carnotgroups with left-invariant sub-Finslerian metric. At the end are describedhomogeneous manifolds such that any invariant inner metric on any ofthem is Finslerian.
Keywords and phrases:
Carnot group, Cohn-Vossen theorem, Gromov-Haudorff limit, homogeneous isotropy irreducible space, homogeneous man-ifold with inner metric, homogeneous space with integrable invariant distri-butions, homogeneous (sub-)Finslerian manifold, homogeneous (sub-)Rie-mannian manifold, Lie algebra, Lie group, locally compact homogeneousgeodesic space, non-holonomic metric geometry, Rashevsky-Chow theorem,shortest arc, submetry, symmetric space, tangent cone.
Introduction
One can observe in last decades an intensive development of non-holonomic met-ric geometry and its applications to geometric group theory, analysis, CR -manifolds,the theory of hypo-elliptic differential equations, non-holonomic mechanics, math-ematical physics, thermodynamics, neurophysiology of vision etc. R.Montgomery’sbook [1] gives a well written track of this. A natural context for (sub-)Finslerian, inparticular, (sub-)Riemannian geometry is geometric control theory [2], [3].Homogeneous Riemannian and Finslerian manifolds and their non-holonomic gen-eralizations, homogeneous sub-Riemannian and sub-Finsleian manifolds, are espe-cially important as models, because in some cases it is possible to find exactlygeodesics, shortest arcs, conjugate and cut locus, and even distances for them. The author was partially supported by the Russian Foundation for Basic Research (Grant 14-01-00068-a) and a grant of the Government of the Russian Federation for the State Support ofScientific Research (Agreement №14.B25.31.0029).
A simple geometric axiomatic for homogeneous (sub-)Finslerian, in particular,(sub-)Riemannian, manifolds in general context of locally compact homogeneousspaces with inner metric, have been announced in paper [4]. Later appeared proofsof this announcement [5], [6], [7], other interpretations, my later results and theirsurvey in [8], corrections to proofs of some results cited in [4]. Also last years somecolleagues exhibited an interest in my old results from [4]. I am very obliged toprofessor D.V.Alekseevsky for useful discussions on this matter. Hopefully, all thisserves as enough motivation to present a modified, renewed, relatively short (withomission of well-known definitions), version of some statements from [4] and [8].1.
Locally compact homogeneous spaces with inner metric
Let us remind main definitions.
A path in a topological space X is a continuousmap of some closed bounded interval of the real line to the space X . A metric spaceis called the space with inner metric , if the distance between any two its points isequal to the infimum of the length of paths joining these points. A metric space is homogeneous if its isometry group acts transitively, i.e., for any two points in thespace there is an isometry (motion) of the space moving one of these points to theother.The S. E. Cohn-Vossen theorem [9] states that every locally compact completespace ( M, ρ ) with inner metric is finitely compact , i.e., any closed bounded subsetin ( M, ρ ) (in particular, any closed ball B ( x, r ) of radius r with the center at x )is compact; moreover, the space ( M, ρ ) is geodesic . The last statement means thatany two points of the space can be joined by a segment or shortest arc , i.e., a curve(path) of length which is equal to the distance between these points.Further we suppose that ( M, ρ ) is an arbitrary locally compact homogeneous spacewith inner metric ρ , and G = I ( M ) is its motion group with compact-open topologywith respect to its action on ( M, ρ ) , G is a connected component of the unit inthe group G. In view of homogeneity and the local compactness, the space ( M, ρ ) is metrically complete, the S. E. Cohn-Vossen theorem holds, and so we can use ashorter term ” locally compact homogeneous geodesic space ”.The Busemann metric , see [10], δ p ( f, g ) = sup x ∈ M ρ ( f ( x ) , g ( x )) e − ρ ( p,x ) , where p ∈ M, is introduced on the group G .The following results are proved in [11]. The metric δ p depends on the choice of thepoint p ∈ M , but it is bi-Lipschitz equivalent to the metric δ q for any point q ∈ M, and thus, independently on the point p ∈ M, defines a topology τ , which coincideswith the compact-open topology on G with respect to its action on ( M, ρ ) . Let usremind that the subbasis of the compact-open topology consists of sets G ( K ; U ) := { g ∈ G | g ( K ) ⊂ U } , where K is a compact and U is an open subset in M . Themetric δ p is invariant under the left translations by elements of the group G andunder the right translations by elements of its compact subgroup H , the stabilizerof the groups G at a point p ; for natural identification of G/H with M, defined byformula σ ( gH ) = g ( p ) , the quotient metric ∆ p on G/H, induced by the metric δ p , is OMOGENEOUS SPACES WITH INNER METRIC 3 equivalent to the metric ρ, the metric space ( G, δ p ) is locally compact, complete, andseparable. The pair ( G, τ ) is a topological group acting continuously and properlyon the left on ( M, ρ ) by isometries. The subgroup G is transitive on M. The following interesting problem is still open.
Problem 1. [12]
Is it true, that in the general case, the connected group G oranother transitive on M closed connected subgroup of the group G is locally connectedor, which is equivalent, locally arcwise connected? Observe in relation with this, that the paper [12] gives a very short proof of a new(at that time) result, that is a global form of a theorem on the local representationof a group as a direct product coming from the Iwasawa-Gleason-Yamabe theory[13, 14, 15] for locally compact groups.
Theorem 1.
Let G be a connected locally compact (Hausdorff ) topological group.Then there exists a compact subgroup K ⊂ G, a connected, simply connected Liegroup L, and a surjective local isomorphism π : K × L → G. Furthermore, if G islocally connected, then K is connected and locally connected, and π is a coveringepimorphism. The following characterization of locally compact homogeneous geodesic spacesas homogeneous spaces of topological groups [12] holds.
Theorem 2.
Every locally compact homogeneous geodesic space is isometric to somelocally compact locally connected quotient space
G/H of a connected locally compacttopological group G with the first countability axiom, by a compact subgroup H, endowed with a G -invariant geodesic metric.Conversely, every locally connected, locally compact homogeneous quotient space G/H of a connected locally compact topological group G with the first countabilityaxiom by a compact subgroup H admits a G -invariant geodesic metric ρ . Corollary 1.
A locally compact topological group ( G, τ ) admits some left-invariantgeodesic metric if and only if ( G, τ ) is connected, locally connected, and satisfies thefirst countability axiom. Theorem 3. [15]
Any neighbourhood U of the unit e in a connected locally compacttopological group G contains closed (even compact) normal subgroups N = N U withthe quotient group G/N, which is a (connected) Lie group.
Lemma 1. [16] If N and N are normal subgroups of a locally compact topologicalgroup G such that G/N and G/N are Lie groups, then G/ ( N ∩ N ) is also a Liegroup. We need the following definition in order to formulate other structural results onlocally compact homogeneous geodesic spaces.
Definition 1. [17]
A map of metric spaces f : M → N is said to be submetry, if forany point x ∈ M , and any number r > , we have f ( B M ( x, r )) = B N ( f ( x ) , r ) . Here B denotes the closed ball of corresponding radius in the corresponding space. V. N. BERESTOVSKII
Theorem 4. [17]
Any Riemannian submersion of complete smooth Riemannianmanifolds is a submetry. Conversely, a submetry of smooth Riemannian manifoldsis the Riemannian submersion of class C , . On the ground of theorems 2, 3, lemma 1, and definition 1, we prove the following
Theorem 5. [4, 6]
A metric space ( M, ρ ) is a locally compact homogeneous geo-desic space, if and only if, it can be represented as the inverse metric limit of somesequence ( M n = ( G/N n ) / ( HN n /N n ) , ρ n ) , where N n is non-increasing sequence ofcompact normal subgroups of G such that ∩ ∞ n =1 N n = { e } , of homogeneous geodesicmanifolds bound by the proper (the preimage of a compact set is compact) submetries p nm : ( M m , ρ m ) → ( M n , ρ n ) , n ≤ m, and p n : ( M, ρ ) → ( M n , ρ n ) , where p n = p nm ◦ p m and p ns = p nm ◦ p ms if n ≤ m ≤ s. This means that (non-decreasing) functions ρ n ◦ ( p n × p n ) uniformly converge tothe metric ρ. Under this condition, p nm ∈ C ∞ , and one can assume that the fibersof these submetries are connected. In some sense, Theorem 5 reduces the study of locally compact homogeneousgeodesic spaces to the case of homogeneous geodesic manifolds.Let us recall, that the
Hausdorff distance d H ( A, B ) between two bounded subsetsof an arbitrary metric space M is the infimum of positive numbers r , such that A is contained in the r -neighbourhood of the set B , and B is contained in the r -neighbourhood of the set A. The pair ( K ( M ) , d H ) is a metric space where K ( M ) isthe family of all closed bounded subsets of the metric space M . It is complete if thespace M is complete [18]. Definition 2.
The Gromov-Hausdorff distance d GH ( A, B ) between compact metricspaces is defined as the infimum of all distances d H ( f ( A ) , g ( B )) for all metric spaces M , and for all isometric embeddings f : A → M, and g : B → M. By definition,a sequence (( X n , x n ) , ρ n ) of finitely compact complete spaces with metrics ρ n andchosen points x n Gromov-Hausdorff-converges to a similar space (( X, x ) , ρ ) , if forany number r > , the distance d GH ( B X n ( x n , r ) , B X ( x, r )) → , as n → + ∞ . Definition 3.
The distance d BGH between finitely compact metric spaces with chosenpoints ( X, x ) and ( Y, y ) is equal by definition to (1) d BGH (( X, x ) , ( Y, y )) = sup r ≥ d GH ( B X ( x, r ) , B Y ( y, r )) e − r . As a consequence of S. E. Cohn-Vossen theorem, cited above, this definition isapplicable to locally compact complete spaces with inner metric, in particular, tolocally compact homogeneous spaces with inner metric. It is clear that in the lattercase the distance d BGH does not depend on the choice of points x ∈ X and y ∈ Y .Let Σ and Θ be respectively the classes of all finitely compact metric spaces andlocally compact complete inner metric spaces with chosen points, and let Ω be aclass of all locally compact homogeneous spaces with inner metric. OMOGENEOUS SPACES WITH INNER METRIC 5
Theorem 6.
The pair (Σ , d BGH ) is a complete metric space. The convergence ofsequences in this metric space is equivalent to the Gromov-Hausdorff convergence.So Θ and Ω are closed subspaces of (Σ , d BGH ) . Moreover, the subclass M of homo-geneous manifolds with inner metric is everywhere dense in (Ω , d BGH ) . Homogeneous manifolds with inner metric
Theorem 7.
The following statements for locally compact homogeneous space withinner metric ( M, ρ ) are equivalent:(1) M is a (connected) topological manifold;(2) M has finite topological dimension;(3) M is locally contractible;(4) ( M, ρ ) is isometric to ( G/H, d ) , where G is a connected Lie group, H is acompact Lie subgroup of G , and d is some inner metric on G/H, invariant relativeto the canonical left action of G on G/H.
Let us give some explanations. Evidently, (1) implies (2) and (3); (4) impliesother statements. Now (2), theorems 2 and 5 imply that M = G/H, where G is alocally compact, connected topological group such that some neighborhood U of e contains no nontrivial normal subgroup, and H is a compact subgroup of G . Thentheorem 3 implies that G is a connected Lie group, H is a compact Lie subgroup of G, which proves (4).The statement (3) together with theorem 2 would imply the statement (4) byJ. Szenthe’s claim in [19]: let a σ -compact locally compact group G, with a compactquotient G/G , acts continuously (and properly) as a transitive and faithful trans-formation group on a locally contractible space X. Then X is a manifold and G isa Lie group. However, it was discovered in 2011 by S. Antonyan [20], that Szenthe’s proof ofthis claim contains a serious gap. Independently Szenthe’s claim was proved byS. Antonyan and T. Dobrowolski [21], by K. H. Hoffmann and L. Kramer [22], seealso the book by K.H.Hoffmann and S.A.Morris [23], pp. 592–605.Theorem 7 gives topological characterization of homogeneous manifolds with innermetric. Now we shall describe their metric structure.Let M = G/H be the quotient manifold of a connected Lie group G by its compactLie subgroup H ; g = G e , h = H e be Lie algebras of Lie groups G, H.
Let us set thefollowing objects:(a1) L e is Ad ( H ) -invariant vector subspace of g such that h ⊂ L e and g is theleast Lie subalgebra of g which contains L e ; (a2) D H = dp ( e )( L ) , where p : G → G/H is the canonical projection and dp isits differential;(a3) F H is a norm on D H which is invariant relative to the (linear) isotropy groupof G/H at H ∈ G/H ; (a4) D is G -invariant distribution on G/H such that D ( H ) = D H ; (a5) F is G -invariant norm on the distribution D such that F ( H ) = F H . Theorem 8. [4] , [5] , [7] Let M = G/H be the quotient space of a connected Liegroup G by its compact Lie subgroup H, ( D, F ) is a pair with conditions (a1)—(a5). V. N. BERESTOVSKII
Then the formula (2) d c ( x, y ) = inf c Z F ( ˙ c ( t )) dt, where c = c ( t ) , ≤ t ≤ , are arbitrary piecewise smooth paths in G/H, tangentto distribution D and joining points x and y from G/H, defines some G -invariantgeodesic metric d c on G/H (compatible with the standard topology on
G/H ). Remark 1.
Conditions (a1), (a2), and (a4) imply that the distribution D fromtheorem 8 is completely nonholonomic [24] . Therefore any two points x and y from G/H can be joined by some piecewise smooth path by Rashevsky-Chow theorem [24] , [25] , so d c ( x, y ) is finite. If D = T M (respectively, D = T M ) then d c is said to be(sub-)Finslerian and (sub-)Riemannian if additionally F H is an Euclidean norm on D H . Note that a norm k · k on a vector space V is Euclidean if and only if k a + b k + k a − b k = 2( k a k + k b k ) for every a, b ∈ V. Theorem 9. [5] , [7] Every G -invariant inner metric ρ on a homogeneous manifold G/H of connected Lie group G by its compact subgroup H is sub-Finslerian orFinslerian. In addition, the Lie group G admits a left-invariant sub-Finslerian orFinslerian (sub-Riemannian or Riemannian if ρ is sub-Riemannian or Riemannian)metric ρ such that the canonical projection p : ( G, ρ ) → ( G/H, ρ ) is submetry. Using this theorem, it is not difficult to prove the following addition to theorem6.
Theorem 10. [8]
The class LG of connected Lie groups with left-invariant Finsle-rian metric is everywhere dense in (Ω , d BGH ) . The last statement of theorem 9 implies that the search of geodesics and short-est arcs of invariant (sub-)Finslerian or (sub-)Riemannian metric on homogeneousmanifolds reduces in many respects to the case of Lie groups with left-invariant(sub-)Finslerian or (sub-)Riemannian metric.Any shortest arc, parametrized by the arc-length, on ( G, ρ ) from theorem 9is a solution of a time-optimal problem ; so it necessarily satisfies the Pontryaginmaximum principle (PMP) [26], [4]. Unfortunately, this principle is useful onlyfor so-called normal shortest arcs and geodesics, when a maximum, supplied forthem by PMP, is positive. Every normal geodesic on ( G, ρ ) is smooth if ρ is sub-Riemannian metric; moreover, if any geodesic on ( G, ρ ) is normal (which is alwaystrue if ρ is Riemannian) then any geodesic on ( G/H, ρ ) is smooth.Let us note that using PMP, the author found in paper [27] all geodesics andshortest arcs of arbitrary left-invariant sub-Finslerian metric on three-dimensionalHeisenberg group. 3. Tangent cones and Carnot groups
Definition 4.
A bijection of metric space ( M, ρ ) onto itself is called a (metric) a -similarity, if ρ ( f ( x ) , f ( y )) = aρ ( x, y ) for all points x, y ∈ M , where a ∈ R , a > .The a -similarity is called nontrivial, if a = 1 . OMOGENEOUS SPACES WITH INNER METRIC 7
Theorem 11. [28] , [29] A locally compact homogeneous space with an inner metric ( M, ρ ) admits nontrivial metric similarities if and only if ( M, ρ ) is isometric to afinite-dimensional normed vector space or to a Carnot group, i.e. connected, simplyconnected, (noncommutative) nilpotent stratified Lie group C with the Lie algebra LC = L = L mk =1 L k (of nilpotentness depth m > ), which is a direct sum of vectorsubspaces L k ⊂ L under the conditions L i +1 = [ L i , L ]; L k = 0 if k > m ; with left-invariant sub-Finslerian metric d cc , defined by a left-invariant norm F c on the left-invariant distribution ∆ , where ∆( e ) = L . Moreover, ( M, ρ ) admits a -similarities for all positive a ∈ R . Theorem 12. [30] If ( M, ρ ) is a homogeneous manifold with inner metric then atany point x ∈ ( M, ρ ) there exists the tangent cone τ x M to the manifold ( M, ρ ) (inthe Gromov’s sense [31] ) as the Gromov-Hausdorff limit of spaces (( M, x ) , αρ ) when α → + ∞ . Let suppose that ( M, ρ ) is ( M = G/H, d c ) as in theorem 8. If d c isFinslerian metric, defined by the norm F on D H = T H M , then τ x M is isometric tothe normed vector space ( T H M, F ); otherwise τ x M is isometric to a Carnot group ( C, d cc ) , where normed vector spaces ( L , F c ) and ( D H , F ) are isometric. Let us note that it follows from theorem 6 and the first statement of theorem 12that τ x M is a locally compact homogeneous space with inner metric which has a -similarities for every positive number a . Now other statements of theorem 12 couldbe deduced from theorem 11.4. Homogeneous Finsler manifolds
Theorem 13. [28]
A metric space ( M, ρ ) is isometric to a homogeneous Finslerianmanifold if and only if ( M, ρ ) is locally compact homogeneous space with inner metricof finite topological dimension which is equal to its Hausdorff dimension.Proof. The necessity of these conditions is well-known.Sufficiency. By theorems 7, 9, every locally compact homogeneous space withinner metric ( M, ρ ) of finite topological dimension is a (sub-)Finslerian homogeneousmanifold, defined by conditions from theorem 8.If F H from theorem 8 is not Euclidean, then we can find an Euclidean norm F H on D H , invariant relative to the (linear) isotropy group of G/H at H ∈ G/H.
Then thereis a constant c > such that (1 /c ) F H ≤ F H ≤ cF H . Now let ρ = d c be G -invariant(sub-)Riemannian metric on G/H defined by formula (2), where F is G -invariantnorm on D such that F ( H ) = F H . Then it is easy to see that (1 /c ) ρ ≤ ρ ≤ cρ and therefore ( M, ρ ) and ( M, ρ ) have equal Hausdorff dimensions.The space ( M, ρ ) is Finslerian if and only if ( M, ρ ) is Riemannian. To finish proofit is enough to apply for ( M, ρ ) theorems 12, 11, and known facts that so-called equiregular connected smooth sub-Riemannian manifold M and any its tangent cone τ x M have equal Hausdorff dimensions, while the Hausdorff dimension of the Carnotgroup ( C, d cc ) from theorem 11 is equal to m X k =1 k dim( L k ) > m X k =1 dim( L k ) = dim( T H ( G/H )) , m > . V. N. BERESTOVSKII (cid:3)
Theorem 14. [4] , [5] Every Lie group with bi-invariant (i.e. with left- and right-invariant) inner metric is the Lie group with bi-invariant Finslerian metric. Homogeneous (sub-)Riemannian manifolds
Theorem 15. [12]
A metric space ( M, ρ ) is isometric to a homogeneous Riemann-ian manifold if and only if ( M, ρ ) is a locally compact homogeneous space with innermetric of finite topological dimension which has the curvature ≥ K in A. D. Alek-sandrov’s sense for some K ∈ R . Notice that there are different equivalent definitions of Aleksandrov spaces ofcurvature ≥ K [32],[33]. The following definition belongs to the author. Definition 5. [34] , [12] A space M with an inner metric ρ and with the local ex-istence of shortest arcs is called the Aleksandrov space of curvature ≥ K , if locallyany quadruple of points in M is isometric to some quadruple of points in a simplyconnected complete Riemannian 3-manifold of some constant sectional curvature k ≥ K, where the number k depends on considered quadruple of points. Remark 2.
There are infinite dimensional compact homogeneous spaces with innermetric of Aleksandrov curvature ≥ [12] . A smooth Riemannian manifold hasAleksandrov curvature ≥ K if and only if its sectional curvatures ≥ K. The definition5 is local, but every quadruple of points in geodesic Aleksandrov space of curvature ≥ K in a sense of this definition satisfies conditions from definition 5 [33] . Some otherconditions, in terms of orbits of 1-parameter subgroups of isometries, characterizinghomogeneous Finsler and Riemannian manifolds, are given in papers [4] , [7] . I don’t know simple metric conditions, characterizing homogeneous sub-Riemannianmanifolds, aside as the Gromov-Hausdorff limits of homogeneous Riemannian man-ifolds, when limits have different finite topological and Hausdorff dimensions.It is interesting that there is a probabilistic approach to solve this problem, atleast in the case of left-invariant inner metrics on Lie groups.
Theorem 16. [35] , [8] Left-invariant (sub-)Riemannian metrics on a connected Liegroup are in 1-1 correspondence with symmetric Gaussian 1-parameter convolutionsemigroups of { e } -continuous, absolutely continuous with respect to left-invariantHaar measure, probability measures on it. Omitting details, we reference to exact definitions and theorem 6.3.8 in book[35]which characterizes generating infinitesimal (hypo-)elliptic operators of such semi-groups. Notice that there is no mention to left-invariant (sub-)Riemannian metricson Lie groups in [35].
Problem 2.
It would be desirable to get a generalization of theorem 6.3.8 in [35] tothe case of homogeneous manifolds
G/H and use it for (sub-)Riemannian geometry.
OMOGENEOUS SPACES WITH INNER METRIC 9 Homogeneous manifolds with integrable invariant distributions
In this section we consider very natural problem: describe connected homogeneousmanifolds such that every invariant inner metric on any of them is Finslerian . Theorem 17. [7] , [4] Every G -invariant inner (geodesic) metric on the homogeneousspace G/H of a connected Lie group G with a compact stabilizer H ⊂ G is Finslerianif and only if(A) Every G -invariant distribution on G/H is integrable.This is equivalent to the condition(B) Every Ad ( H ) -invariant vector subspace c in g , containing h , is a Lie algebra.If H is connected, in particular, if G/H is simply connected, then the Ad ( H ) -invariance of the space c is equivalent to the inclusion [ h , c ] ⊂ c. Definition 6.
The homogeneous manifold
G/H of a connected Lie group G witha compact stabilizer H is called homogeneous manifold with integrable invariantdistributions, shortly, HMIID, if it satisfies any of the equivalent conditions (A) or(B) from theorem 17. Theorem 18. [7] , [4] The following conditions for a connected Lie group G withthe Lie algebra g are equivalent: Every left-invariant inner metric on the Lie group G is Finslerian; Every vector subspace of the Lie algebra g is a Lie subalgebra in g ; g is one-dimensional or any two-dimensional vector subspace in g is a Liesubalgebra of g ; For any two elements
X, Y in g , the bracket [ X, Y ] is a linear combinationof elements X and Y. Theorem 19. [36]
If a Lie algebra g satisfies condition 4) from theorem 18 then there exists a linear map l : g → R such that (3) [ X, Y ] = l ( X ) Y − l ( Y ) X, X, Y ∈ g ; the kernel of the linear map l is the maximal commutative ideal in g ; l = 0 if and only if g is commutative Lie algebra; if l = 0 then, up to an isomorphism, the Lie algebra g has the form L n = R + φ R n − , n ≥ , i.e. semidirect sum, prescribed by homomorphism φ : R → End ( R n − ) , suchthat φ (1) = E n − is the unit matrix. Theorem 20. [36]
Let G n be n -dimensional Lie group G with the Lie algebra g ,satisfying condition 4) from theorem 18. Then G n is commutative or G n is isomorphic to the group of real ( n × n ) block matrices (4) (cid:18) aE n − b (cid:19) , where E n − is the unit ( n − × ( n − -matrix, a is any positive number, b is any ( n − -vector-column, and is the zero ( n − -vector-row. Theorem 21. [36]
Noncommutative Lie group G has the Lie algebra g satisfyingcondition 4) from theorem 18 if and only if any left-invariant Riemannian metricon G has constant negative curvature. Theorem 22. [7] , [4] Let M be a connected Riemannian symmetric space, G be themaximal connected Lie group of isometries for M with the stabilizer H ⊂ G at apoint x ∈ M. Then any G -invariant inner metric on G/H is Finslerian.
Theorem 23. [7] , [4] Let assume that M = G/H (where G is a connected Liegroup and H is its compact subgroup) be isotropy irreducible homogeneous spaces,i.e. G/H has irreducible linear isotropy group. Then any G -invariant inner metricon G/H is Finslerian.
Theorem 24. [37] , [38] , [39] For any (compact) simply connected effective homoge-neous space
G/H of a connected compact Lie group G with closed stabilizer H thefollowing conditions are equivalent: all G -invariant distributions on G/H are integrable; the homogeneous space G/H is isomorphic to a direct product of compactsimply connected isotropy irreducible homogeneous spaces; the space G/H has normal type by Berard-Bergery [40] , i.e. any G -invariantRiemannian metric on G/H is normal homogeneous in M.Berger’s sense; all G -invariant Riemannian metrics on G/H have positive Ricci curvature; all G -invariant Riemannian metrics on G/H have positive scalar curvature.
Simply connected irreducible (Riemannian) symmetric spaces
G/H with con-nected Lie group G and compact subgroup H have been classified by E Cartanin 1926, see [41]. They are (automatically) strictly isotropy irreducible, i.e. H hasirreducible isotropy representation; non-compact strictly isotropy irreducible homo-geneous spaces are symmetric [42]. O. Manturov [42] and J. A. Wolf [43] classifiedstrictly isotropy irreducible homogeneous spaces; one needs to combine their resultsto get full classification; see also [41].The author used no classification when he proved his results stated in this paper.It follows from previous statements that there is a full classification of compactsimply connected HMIID.V. V. Gorbatsevich [44], [8] studied general homogeneous spaces with connectedstabilizer subgroup from the class HMIID in detail. He described correspondingtransitive Lie groups and stabilizer subgroups in the case when the transitive groupis semisimple or solvable, and partly, in the case of general transitive Lie groups. References [1]
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Sobolev Institute of Mathematics SD RAS, NovosibirskV.N.Berestovskii
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