Einstein Homogeneous Riemannian Fibrations

Transcription

1 Einstein Homogeneous Riemannian Fibrations Fátima Araújo Doctor of Philosophy University of Edinburgh 008

2 To my godparents Amadeu and Conceição

3 Acknowledgements I owe my deepest gratitute to my supervisor Prof. Dmitri Alekseevskii, who has invested considerable time and energy into guiding me through my PhD. His professional guidance and unfaltering support have been most invaluable. I would like to thank Prof. Elmer Rees for his supervision in a first stage of my PhD and Dr. Roger Bielawski for his help and ideas during a critical period of my research. I would like to thank the Fundação Portuguesa para a Ciência e Tecnologia for sponsoring this work and the Centro de Algebra da Universidade de Lisboa. Last but not least, I want to thank my family and close friends for their unwavering emotional support and firm belief in me throughout my research and life.

4 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Fátima Araújo)

5 Abstract This thesis is dedicated to the study of the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for the existence of Einstein metrics with totally geodesic fibers in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We investigate the existence of such Einstein metrics for invariant bisymmetric fibrations of maximal rank, i.e., when both the base and the fiber are symmetric spaces and the base is an isotropy irreducible space of maximal rank. We find this way new Einstein metrics. For such spaces we describe explicitly the isotropy representation in terms subsets of roots and compute the eigenvalues of the Casimir operators of the fiber along the horizontal direction. Results for compact simply connected 4-symmetric spaces of maximal rank follow from this. Also, new invariant Einstein metrics are found on Kowalski n-symmetric spaces.

6 Table of Contents Chapter 3. The Ricci Curvature of a Riemannian Homogeneous Space Homogeneous Riemannian Fibrations Notation and Hypothesis The Ricci Curvature in the Direction of the Fiber The Ricci Curvature in the Horizontal Direction The Ricci Curvature in the Mixed Direction Necessary Conditions for the Existence of an Adapted Einstein Metric Chapter 9. Riemannian Fibrations with Normal Fiber Riemannian Fibrations with Standard Base Binormal Riemannian Fibrations Riemannian Fibrations with Einstein Fiber and Einstein Base Riemannian Fibrations with Symmetric Fiber Chapter Introduction Bisymmetric Triples of Maximal Rank - Classification Einstein Adapted Metrics for Type I Einstein Adapted Metrics for Type II Application to 4-symmetric Spaces Tables Chapter Kowalski N-Symmetric Spaces - The Isotropy Representation and the Casimir Operators Existence of Einstein Adapted Metrics Closing Remarks Appendix A 00 A. A n

7 A. B n A.3 D n A.4 C n A.5 f A.6 g A.7 e A.8 e A.9 e Bibliography 36

8 Introduction The principal topic of study in this thesis is the existence of Einstein invariant metrics on the total space of homogeneous Riemannian fibrations such that the fibers are totally geodesic submanifolds. In chapter we introduce some main definitions and notation and deduce some essential formulas for the Ricci curvature of an invariant metric. Then we consider a fibration G/L G/K such that G is a compact connected semisimple Lie group G, and L K G are connected closed non-trivial subgroups of G. We assume that G/L has simple spectrum. On the total space G/L, we consider a G-invariant Riemannian metric such that the natural projection G/L G/K is a Riemannian submersion and the fibers are totally geodesic submanifolds. We shall briefly call such a metric an Einstein adapted metric. We describe the Ricci curvature of any adapted metric in terms of Casimir operators and obtain two necessary conditions for existence of an Einstein adapted metric expressed only in terms of the Casimir operators of the horizontal and vertical directions. These will provide a tool to show that in many cases such a metric cannot exist without further study, i.e., only by studying eigenvalues of certain Casimir operators. In chapter we restrict the object of study to some special cases, where the Einstein equations are simpler. We consider the cases where the metric on the fiber or on the base is a multiple of the Killing form of G, in which cases are included those with isotropy irreducible fiber or base. The case when both these two conditions are satisfied gives rise to the study of the, throughout called, Einstein binormal metrics. The existence of Einstein binormal metrics translates into very simple algebraic conditions which shall allow us to find out new Einstein metrics. Also we obtain necessary conditions for existence of an Einstein adapted metric such that the metric on the base space or the metric on the fiber are also Einstein. Finally, we apply the results obtained so far to the case when the fiber is a symmetric space and N is isotropy irreducible. Chapter 3 is devoted to bisymmetric fibrations of maximal rank, i.e., we consider a fibration G/L G/K, as in chapter, such that L is a subgroup of maximal rank, K/L is a symmetric space and G/L is an isotropy irreducible symmetric space. We introduce the notion of a bisymmetric triple (g, k, l) associated to a bisymmetric fibration. We obtain all the bisymmetric triples (g, k, l) in the case

9 when g is a simple Lie algebra and classify them into two different types, I and II. We classify all the Einstein adapted metrics when g is an exceptional Lie algebra, for both Type I and II. When g is a classical Lie algebra, we classify all the Einstein adapted metrics for Type I. For Type II in the classical case, we classify all Einstein binormal metrics and all Einstein adapted metrics whose restriction to the fiber is also Einstein; moreover, if one of these metrics exists we obtain all the other Einstein adapted metrics. Finally, we apply the results obtained to compact simply-connected irreducible 4-symmetric spaces. In appendix A we obtain all the necessary eigenvalues for the Einstein equations for each bisymmetric triple considered in this chapter. In chapter 4 we study the existence of Einstein adapted metrics on the Kowalski n-symmetric spaces, i.e., we consider a fibration p G 0 q G 0 n G 0 Gn 0 n G 0 Gp 0 p G 0 Gq 0 q G 0, where G 0 is compact connected simple Lie group and m G 0 is the diagonal subgroup in G m 0, for m = p, q, n. It is known that G n 0 n G 0 is a standard Einstein manifold and we prove that, for n > 4, there exists another Einstein adapted metric, whereas, for n = 4, the standard metric is the only Einstein adapted metric.

10 CHAPTER In Section we introduce some essential definitions and notation. We deduce a formula for the Ricci curvature of an invariant metric on a reductive homogeneous space. In Section we obtain the Ricci curvature of an invariant metric with totally geodesic fibers on the total space of a homogeneous fibration and some necessary conditions for that metric to be Einstein.. The Ricci Curvature of a Riemannian Homogeneous Space A Riemannian metric g is said to be Einstein if its Ricci curvature satisfies an equation of the form Ric = Eg, for some constant E, the Einstein constant of g ([0]). This equation, commonly called the Einstein equation, is in general a very complicated system of partial differential equations of second order. Although so far no fully general results are known for existence of Einstein metrics, many results of existence and classification are known for many classes of spaces. Two examples of this are the Kähler-Einstein metrics ([49], [5], [40], [43]) and the Sasakian-Einstein metrics ([]). Many results are known on homogeneous Einstein metrics. For Riemannian homogeneous spaces the Einstein equation translates into a system of algebraic equations, which is an easier problem than its general version. However, even for this class of spaces we are far from knowing full answers. Einstein normal homogeneous manifolds were classified by Wang and Ziller ([44]). Nowadays, it is known that every compact simply connected homogeneous manifold with dimension less or equal to admits a homogeneous Einstein metric: any such manifold with dimension or 3 has constant sectional curvature [0]; in dimension 4, the result was shown by Jensen ([8]), and by Alekseevsky, Dotti and Ferraris in dimension 5 ([4]); in dimension 6, the result is due to Nikonorov and Rodionov ([3]), and in dimension 7 it is due to Castellani, Romans and Warner ([4]). All the 7-dimensional homogeneous Einstein manifolds ([9]) were obtained by Nikonorov. These results were extended to dimension up to by Böhm and Kerr ([]). Many examples of homogeneous Einstein manifolds with dimension arbitrary big are known. Spheres and projective spaces 3

11 are examples of this, where all homogeneous Einstein metrics were classified by Ziller ([5]). Also isotropy irreducible spaces ([47]), symmetric spaces ([6],[]), flag manifolds, among many others, provide examples of Einstein manifolds with arbitrary big dimension. Einstein homogeneous fibrations have also been the object of study. We recall the work of Jensen on principal fibers bundles ([9]), where new invariant Einstein metrics are found on the total space of certain homogeneous fibrations, and the work of Wang and Ziller on principal torus bundles ([46]). Einstein homogeneous fibrations are the main focus of this thesis. Let G be a Lie group and L a closed subgroup. We denote by g and l the Lie algebras of G and L, respectively. The homogeneous space M = G/L is the space of all cosets {al : a G} endowed with the unique differentiable structure such that the canonical projection π : G M a al (.) is a submersion, i.e., dπ a is onto for all a G, and with the natural transitive left action of G, τ : G M M (b, al) (ba)l (.) Let X g and let exp tx be the one-parameter subgroup generated by X. For every a G, dπ a (X) = d dt (exp tx)al t=0 (.3) and, in particular, for o = π(e) = L, this map yields an isomorphism dπ e : g/l = T o M. (.4) For every X g we define a G-invariant vector field on M by X al = dπ a (X) = d dt (exp tx)gl t=0. (.5) The homogeneous space M is called reductive if there exists a direct complement m of l in g which is Ad L-invariant, i.e., The inclusion Ad L(m) m implies g = l m and Ad L(m) m. (.6) [l, m] m (.7) 4

12 and the converse holds if L is connected. The homogeneous space M is reductive if L is compact. Throughout, we suppose that L is compact and we denote by m a reductive complement of l on g. If M is reductive we have an isomorphism m = T o M and the tangent space T o M is identified with m and consequently the vector field X on M is identified with X m. Under this identification we shall simply write X for Xo. Furthermore, the isotropy representation of M, Ad M : L GL(T o M), is equivalent to the adjoint representation of L on m. Consequently, there is a oneto-one correspondence between G-invariant objects on M and Ad L-invariant objects on m. In particular, G-invariant metrics on M correspond to Ad L-invariant scalar products on m. More precisely, a metric g on M is said to be G-invariant if, for every a G, τ a g = g and the correspondence between G-invariant metrics on M and Ad L-scalar products on m is given g a (X a, Y a ) =< X, Y >, for all a G. (.8) Let Kill be the Killing form of g. We recall that Kill is the bilinear form on g defined by Kill(X, Y ) = tr(ad X ad Y ), X, Y g (.9) where, for each X g, ad X denotes the adjoint map g g Y [X, Y ]. (.0) The Killing form of g is an Ad G-invariant bilinear form and it is non-degenerate if g is semisimple. Moreover, if G is a compact connected semisimple Lie group, Kill is negative definite. In this case, by (.8), the negative of the Killing form induces a G-invariant metric on M, the standard Riemannian metric on M. With respect to the decomposition g = l m, we write ( 0 CX ad X = B X P X ), for every X m. (.) 5

13 Hence, for Y m, (ad X Y ) m = P X Y and (ad XY ) m = (B X C X + PX)Y, (.) where the subscript m denotes the projection onto m. Let g M be a G-invariant metric on M. As was explained above, there is a one-toone correspondence between G-invariant metrics on M and Ad L-invariant scalar products on m. So let <, > be the Ad L-invariant scalar product on m corresponding to g M. For every X m, let T X be the endomorphism of m defined by < T X Y, Z >=< X, P Y Z >, for every Y, Z m. (.3) The Nomizu operator of the scalar product <, > (cf. [8], [5]) is L X End(m), X g, defined by L X Y = Y X, Y m (.4) where is the Riemannian connection of g M. We have where U : m m m is the operator L X Y = P XY + U(X, Y ), (.5) U(X, Y ) = (T XY + T Y X), X, Y m. (.6) The metric g M is called naturally reductive if U = 0. The curvature tensor, the sectional curvature and the Ricci curvature of g M are G-invariant tensors and thus they are determined by the following identities ([8]), which represent their values at the point o. By using L the curvature tensor of g M at o can be written as R(X, Y ) = [L X, L Y ] m L [X,Y ]m ad [X,Y ]l, (.7) for every X, Y m. The sectional curvature K of g M is defined by K(Z, X) =< R(Z, X)X, Z >, (.8) for every X, Z m orthonormal with respect to <, >. The Ricci curvature of g M is determined by 6

14 Ric(X, X) = i K(Z i, X), X m (.9) where (Z i ) i is an orthonormal basis of m with respect to <, >. The metric g M is said to be an Einstein metric if Ric = Eg M, (.0) for some constant E called the Einstein constant of g M. Equipped with a G- invariant Einstein metric, M is called an Einstein homogeneous manifold. Below we show some elementary properties of the Nomizu operator: Lemma.. (i) L X is skew-symmetric with respect to <, >, i.e., (ii) for every X, Y m, < L X Y, Z > + < Y, L X Z >= 0, X, Y, m; (.) L X Y L Y X = [X, Y ] m = P X Y. (.) Proof: From identities (.3), (.5) and (.6) we deduce < L X Y, Z >= < P X Y, Z > < T X Y, Z > < T Y X, Z > = < T Z X, Y > + < T X Z, Y > < P X Z, Y >= = < Y, L X Z > and this shows the skew-symmetry of L X. To show (ii) we just observe that U(X, Y ) = U(Y, X) and P X Y = P Y X. Equivalently, this assertion just means that the Levi-Civita connection is torsion free since X Y = L X Y. For every X, Y m, we define the operator and the vector R X Y = L Y X (.3) Lemma.. For every X m, V X = L X X. (.4) R X = (P X + P X + T X ) and R X = (P X + P X T X ). 7

15 Proof: Let X, Y, Z m. < R X Y, Z >= < L Y X, Z > = < [Y, X] m, Z > + < Y, [Z, X] m, > + < X, [Y, Z] m > = < P X Y, Z > < PX Y, Z > < T XY, Z >. Thus we obtain the required expression for R X. The formula for RX follows from the fact that P X + PX is symmetric and T X is skew symmetric with respect to <, >. Lemma.3. The sectional curvature of g M is K(Z, X) =< ( ) RXR X PX PXP X B X C X + P VX Z, Z >, for every Z, X m orthonormal with respect to <, >. Proof: Let Z, X m. The proof is a straightforward calculation by using the identities (.7), (.) and (.): K(Z, X) = < R(Z, X)X, Z > = < [L Z, L X ] m X, Z > < L [Z,X]m X, Z > < ad [Z,X]l X, Z > = < L Z L X X, Z > < L X L Z X, Z > < L X [Z, X] m, Z > < [[Z, X] m, X] m, Z > < [[Z, X] l X] m, Z > = < L X X, L Z Z, > + < L Z X, L X Z > + < [Z, X] m, L X Z > < [[Z, X], X] m, Z > = < L X X, L Z Z, > + < L Z X, L Z X > + < L Z X, [X, Z] m > + + < [Z, X] m, L Z X > + < [Z, X] m, [X, Z] m > < [X, [X, Z]] m, Z > = < L X X, L Z Z, > + < L Z X, L Z X > < [Z, X] m, [Z, X] m > < (PX Z) m, Z > 8

16 = < L Z X, L Z X > < [X, Z] m, [X, Z] m > < (PX Z) m, Z > < L Z V X, Z > = < R X Z, R X Z > < P X Z, P X Z > < (PX + B XC X )Z, Z > < L VX Z + [Z, V X ] m, Z > = < (RX R X PX P X B X C X PX )Z, Z > + < (P V X L VX )Z, Z >. Since L X is skew-symmetric with respect to <, >, we have < L VX Z, Z >= 0, for every Z m, and we obtain K(Z, X) =< ( ) RXR X PXP X B X C X PX + P VX Z, Z >, as required. Theorem.. Let X, Y m. Then In particular, Ric(X, Y ) = tr(r XR Y + B X C Y + B Y C X P U(X,Y ) ). Ric(X, X) = tr(r X + B X C X P VX ). Proof: We first compute Ric(X, X) and then obtain Ric(X, Y ) by polarization. Since T X is a skew-symmetric operator and P X + P X is symmetric, tr ( (P X + P X)T X ) = tr ( TX (P X + P X) ) = tr ( (P X + P X)T X ) and thus all the terms vanish. Therefore, by using Lemma. we obtain and tr(r XR X ) = 4( (PX + P X) T X ) = 4 tr(p X + P XP X T X) tr(r X) = 4( (PX + P X) + T X Hence, tr(r X R X P X P X P X) = tr(r X ). ) = 4 tr(p X + P XP X + T X). Let us suppose that < X, X >= and let {e i } i be an orthonormal basis for m with respect to <, > such that X = e. Hence, we can apply Lemma.3 to obtain the following: 9

17 Ric(X, X) = i K(e i, X) = < ( RX R X PX P X B X C X PX + P ) V X ei, e i > = tr ( RX R X PX P X P ) X B X C X + P VX = tr(rx + B XC X P VX ). Since both Ric and the map X tr(rx + B XC X P VX ) are bilinear maps, the identity above holds even if X is not unit. Hence, for every X m, Ric(X, X) = tr(rx + B X C X P VX ). Now we compute Ric(X, Y ). Since Ric is a symmetric bilinear operator we have Ric(X, Y ) = Ric(X + Y, X + Y ) Ric(X, X) Ric(Y, Y ). By using the expression above for Ric(X, X) we get Ric(X, Y ) = tr( RX+Y + R X + R Y )+ + tr( B X+Y C X+Y + B X C X + B Y C Y )+ + tr(p VX+Y P VX P VY ). By bilinearity of L and property (.) we obtain V X+Y = L X+Y (X + Y ) = L X X + L Y Y + L X Y + L Y X = V X + V Y + L X Y [X, Y ] m = V X + V Y + L X Y + P X Y. Therefore, P VX+Y P VX P VY = P LX Y P X Y. Also the identity B X+Y C X+Y = B X C X + B Y C Y + B X C Y + B Y C X implies that B X+Y C X+Y + B X C X + B Y C Y = B X C Y B Y C X. Moreover, R X+Y = R X + R Y + R XR Y + R Y R X and thus tr( R X+Y + R X + R Y ) = tr(r X R Y + R Y R X ) = tr(r X R Y ). 0

18 Therefore, we obtain Ric(X, Y ) = tr(r X R Y + B X C Y + B Y C X ) + tr(p LX Y P X Y ) = tr(r X R Y + B X C Y + B Y C X ) + tr(p U(X,Y ) ). Corollary.. Let X, Y m. Ric(X, Y ) = 4 tr(p XP Y + T X T Y ) Kill(X, Y ) + tr(p U(X,Y )). Proof: By using Theorem. and Lemma., we write Ric as follows: Ric(X, Y ) = ( ) (P X + PX + T X )(P Y + PY + T Y ) + B X C Y + B Y C X P U(X,Y ). (.5) Since P X + P X and P Y + P Y are symmetric linear maps and T X and T Y are skew-symmetric, we have Moreover, tr((p X + P X)T Y ) = tr(t X (P Y + P Y )) = 0. (.6) tr(p X P Y ) = tr(p XP Y )) and tr(p XP Y ) = tr(p X P Y )). (.7) We can use (.) to write Kill as follows: Kill(X, Y ) = tr(ad X ad Y ) = tr(c X B Y + B X C Y + P X P Y ). (.8) Finally, by using (.6), (.7) and (.8) we simplify (.5) to obtain the expression stated for Ric. Definition.. A symmetric bilinear map β on m is said to be associative if β([u, v] m, w) = β(u, [v, w] m ), for every u, v, w m. Remark.. If there exists on m an associative symmetric bilinear form β such that β is non-degenerate, then tr(p U(X,Y ) ) = 0, for all X, Y m. Indeed, if such a bilinear form exists, tr P a = 0, for every a m. Let {w i } i and {w i} i be bases of m dual with respect to β, i.e., β(w i, w j) = δ ij. Then, for every a m, β(p a w i, w i) = β([a, w i ] m, w i) = β(w i, [a, w i] m ) = β(p a w i, w i ).

19 Hence, tr(p a ) = 0. Also, if the metric g M on M is naturally reductive, then P U(X,Y ) = 0, for all X, Y m, since, in this case, U is identically zero. Definition.. Let U, V be Ad L-invariant vector subspaces of g. We define a bilinear map Q UV : m m R by Q UV XY = tr([x, [Y, ] V ] U ), X, Y m, where the subscripts U and V denote the projections onto U and V, respectively. Definition.3. Let U be an Ad L-invariant vector subspace of g such that the restriction of Kill to U is non-degenerate. The Casimir operator of U with respect to the Killing form of g is the operator C U = i ad ui ad u i, where {u i } i and {u i} i are bases of U which are dual with respect to Kill, i.e., Kill(u i, u j) = δ ij. The Casimir operator C U is an Ad L-invariant linear map and thus it is scalar on any irreducible Ad L-module. In particular, if g is simple, then C g = Id g. Lemma.4. Suppose that g is semisimple. Let U, V be Ad L-invariant vector subspaces of g such that the restrictions of the Killing form to U and V are both non degenerate. (i) Q UV is an Ad L-invariant symmetric bilinear map. Hence, if W g is any irreducible Ad L-submodule, then Q UV W W is a multiple of Kill W W. (ii) Q UV = Q V U. Proof: Since g is a semisimple Lie algebra its Killing form is non-degenerate. As in addition Kill U U and Kill V V are non-degenerate, we may consider the orthogonal complements U and V of U and V, respectively, in g with respect to Kill. It follows that and Kill U g = Kill(, p U ) U g Kill V g = Kill(, p V ) V g, where p U and p V are the projections onto U and V, respectively.

20 Also, we may consider bases {w i } i and {w i} i of U which are dual with respect to Kill. By using these facts we have the following: (i) Let X, Y m and g L. Kill([X, [Y, w i ] V ] U, w i) = Kill([X, [Y, w i ] V ], w i) = Kill([Y, w i ] V, [X, w i]) = Kill([Y, w i ], [X, w i] V ) = Kill(w i, [Y, [X, w i] V ]) = Kill(w i, [Y, [X, w i] V ] U ). Therefore, tr([x, [Y, ] V ] U ) = tr([y, [X, ] V ] U ) and thus Q UV XY symmetric. = QUV Y X. So QUV is To show the Ad L-invariance of Q UV we note that since V and V are Ad L- invariant subspaces and g = V V, the projections on V and V are also Ad L-invariant linear maps. Kill([Ad g X, [Ad g Y, w i ] V ] U, w i) = Kill([Ad g X, [Ad g Y, w i ] V ], w i) = Kill(Ad g [Ad g X, [Ad g Y, w i ] V ], Ad g w i) = Kill([X, Ad g [Ad g Y, w i ] V ], Ad g w i) = Kill([X, [Y, Ad g w i ] V ], Ad g w i) = Kill([X, [Y, Ad g w i ] V ] U, Ad g w i). If {w i } i and {w i} i are dual bases of U with respect to Kill, then {Ad g w i } i and {Ad g w i} i are still dual bases as well since the Killing form is invariant under inner automorphisms. So by the above we conclude that tr([ad g X, [Ad g Y, ] V ] U ) = tr([x, [Y, ] V ] U ) and thus Q UV is Ad L-invariant. (ii) For Z m, let A Z = (ad Z U ) V and B Z = (ad Z V ) U. We have Q V U XY = tr(a X B Y ) = tr(b Y A X ) = Q UV Y X. Hence, by symmetry of Q UV, we conclude that Q V XY U X, Y m. Therefore, Q UV = Q V U. 3 = QUV Y X = QUV XY, for every

21 Lemma.5. Suppose that g is semisimple. Let U, V be Ad L-invariant vector subspaces of g such that the restrictions of the Killing form to U and V are both non-degenerate. For every X, Y m, (i) if ad X U V or ad Y U V, then Q UV XY = Kill(C UX, Y ) = Kill(X, C U Y ); (ii) if ad X V U or ad Y V U, then Q UV XY = 0; (iv) if ad X ad Y U U or ad Y ad X U U, then Q UV XY = 0. Proof: (i) Let C U = i ad w i ad w i be the Casimir operator of U. Since Q UV = Q V U it suffices to suppose that ad Y U V. If ad Y U V, then Since Q UV XY = tr([x, [Y, ]] U ) = tr(ad X ad Y U ). Kill([X, [Y, w i ]] U, w i) = Kill([X, [Y, w i ]], w i) = Kill([Y, w i ], [X, w i]) = Kill(Y, [w i, [w i, X]]), we have Q UV XY = i Kill(Y, [w i, [w i, X]]) = Kill(Y, C U X). By symmetry of Q UV we also get Q UV XY = Kill(X, C UY ). (ii) If ad X V U, then, for every w, w U, Kill([X, [Y, w] V ], w ) = 0 and thus Q UV XY = 0, for every Y m. By symmetry, the same conclusion holds if ad Y V U. (iii) If ad X ad Y U U, then, for every w, w U, Kill([X, [Y, w]], w ) = 0 and thus Kill([X, [Y, w] V ] U, w ) = 0. Hence Q UV XY = 0. If ad Y ad X U U, then Q UV XY = 0 by symmetry. Remark.. In Lemmas.4 and.5 the condition that g is semisimple may be replaced by requiring that there is on g a non-degenerate Ad L-invariant symmetric bilinear form β, since in the proofs above the Killing form may be replaced by any such form β. In this case, the orthogonality conditions in Lemma.5 should be understood as conditions with respect to β. Theorem.. Let β be an associative Ad L-invariant non-degenerate symmetric bilinear form on m. Let m = m... m m be a decomposition of m into Ad L- invariant subspaces such that β mj m i = 0, if i j. Let g M be the G-invariant pseudo-riemannian metric on G/L induced by the scalar product of the form <, >= m j=ν j β mj m j, (.9) 4

22 for ν j > 0, for every j =,..., m. For every X m a and Y m b, the Ricci curvature of g M is given as follows: Ric(X, Y ) = m j,k= ( νk ν ) aν b Q m jm k XY Kill(X, Y ). ν j ν k ν j Proof: First we observe that the non-degeneracy of β and the condition of pairwise orthogonality of the m j s imply that β mj m j is in fact non-degenerate. Let X m a and Y m b. By Corollary. we have Ric(X, Y ) = 4 tr(p XP Y + T X T Y ) Kill(X, Y ) + tr(p U(X,Y )). According to Remark., we have tr(p U(X,Y ) ) = 0. Let j =,..., m and let {w i } i and {w i} i be dual bases for m j with respect to β. We note that such bases exist as β mj m j < T X T Y w i, w i >= < X, [T Y w i, w i] m > = ν a ν b m k= = ν a ν b m k= = ν a ν b m k= = ν a ν b m k= = ν a ν b m k= = ν a ν b m k= = ν a β(x, [T Y w i, w i]) is non-degenerate. = ν a β(t Y w i, [X, w i]) = ν a m k= β(t Y w i, [X, w i] mk ) = ν a m k= = ν a m k= ν k < T Y w i, [X, w i] mk > ν k < Y, [w i, [X, w i] mk ] m > ν k β(y, [w i, [X, w i] mk ]) ν k β([y, w i ], [X, w i] mk ) ν k β([y, w i ] mk, [X, w i]) ν k β(w i, [X, [Y, w i ] mk ]) ν k β(w i, [X, [Y, w i ] mk ] mj ) ν k ν j < w i, [X, [Y, w i ] mk ] mj >. 5

23 Hence, and thus Then tr(t X T Y mj ) = ν a ν b and thus we get m k= ν k ν j tr([x, [Y, ] mk ] mj ) = ν a ν b tr(t X T Y ) = ν a ν b m j,k= < P X P Y w i, w i >= < P Y w i, P X w i > tr(p XP Y mj ) = ν k ν j Q m jm k XY. m k= = m k= < [Y, w i] mk, [X, w i] mk > = m k= ν kβ([y, w i ] mk, [X, w i]) = m k= ν kβ(w i, [X, [Y, w i ] mk ]) = m k= ν kβ(w i, [X, [Y, w i ] mk ] mj ) = m ν k k= < w ν i, [X, [Y, w i ] mk ] mj >. j m k= ν k ν j tr([x, [Y, ] mk ] mj ) = tr(p XP Y ) = m j,k= ν k ν j Q m jm k XY. m k= Q m jm k XY ν k ν j ν k Q m jm k XY ν j By using Corollary. we finally obtain the required expression for Ric(X, Y ). We recall that a metric g M is said to be normal if it is a multiple of an associative Ad L-invariant non-degenerate symmetric bilinear form on m. Corollary.. If g M is a normal metric, then Ric(m i, m j ) = 0, for all i j. For every X m j, Ric(X, X) = 4 Kill(X, X) Kill(C lx, X), where C l is the Casimir operator of l with respect to the Killing form. Furthermore, if m j is irreducible, then Ric mj m j = ( ) + c l,j Kill mj m j, where c l,j is the eigenvalue of C l on m j. 6

24 Proof: Let X m a and Y m b. If g M is a normal metric, then there exists an Ad L-invariant non-degenerate symmetric bilinear form β on m which induces g M. Hence, in Theorem. we can take ν =... = ν m = and obtain the following: Ric(X, Y ) = 4 m j,k= Qm jm k XY Kill(X, Y ) = 4 Qmm XY Kill(X, Y ) = 4 Qmg XY 4 Qml XY Kill(X, Y ) = 4 Qmg XY 4 Qlm XY Kill(X, Y ) = 4 Kill(C mx, Y ) 4 Kill(C lx, Y ) Kill(X, Y ) = 4 Kill(X, Y ) Kill(C lx, Y ). Since C l (m a ) m a, it is clear that Ric(X, Y ) = 0 if a b and Ric is well determined by elements Ric(X, X) with X m a. If m j is irreducible, then C l is scalar on m j and we obtain the identity given for Ric. The formula above for the Ricci curvature of a normal metric was first found by M.Y. Wang and W. Ziller in [44]. From Corollary., it is clear that a necessary and sufficient condition for a normal metric to be Einstein is that the Casimir operator of l is scalar on the isotropy space m. For instance, this condition holds if m is irreducible. Simply connected non-strongly isotropy irreducible homogeneous spaces which admit a normal Einstein metric were classified by M.Y. Wang and W. Ziller in [44], when G is a compact connected simple group. Also, more generally, simply connected compact standard homogeneous manifolds were studied by E.D. Rodionov in [39]. We obtain a similar formula to that of Corollary., in the case when the submodules m,..., m m pairwise commute. Corollary.3. If m,..., m m pairwise commute, i.e., [m a, m b ] = 0, for all a b, then Ric(m a, m b ) = 0, for all a b. For every X m a, Ric(X, X) = 4 Kill(X, X) Kill(C lx, X), where C l is the Casimir operator of l with respect to the Killing form. Furthermore, if m j is irreducible, then Ric mj m j = where c l,j is the eigenvalue of C l on m j. ( ) + c l,j Kill mj m j, 7

25 Proof: Let X m a and Y m b. If m,..., m m pairwise commute, then, by Lemma.5, we have Q m jm k XY = 0, for every j, k a, b. In particular, all these bilinear maps vanish if a b. Hence, if a b, Ric(X, Y ) = Kill(X, Y ) = 0. Therefore, the Ricci curvature is well determined by elements of the form Ric(X, X), with X m a, and Ric(X, X) = 4 Qmama XX Kill(X, X). Since Q mama XX = Qmm XX, the rest of the proof is similar to the proof of Corollary.. 8

26 . Homogeneous Riemannian Fibrations.. Notation and Hypothesis In this Section we obtain the Ricci curvature of an invariant metric with totally geodesic fibers on the total space of a homogeneous fibration. We start by settling once and for all the notation used throughout. Let G be a compact connected semisimple Lie group and L K G connected closed non-trivial subgroups of G. We denote M = G/L, N = G/K and F = K/L. We consider the natural fibration π : M N al ak (.30) with fiber F and structural group K. We denote by g, k and l the Lie algebras of G, K and L, respectively. By Kill we denote the Killing form of G and we set B = Kill. Also, we denote the Killing forms of K and L by Kill k and Kill l, respectively. As G is compact and semisimple, the Killing form of G is negative definite and thus B is positive definite. We consider an orthogonal decomposition of g with respect to B given by where k = l p. g = l p n, (.3) }{{} m Clearly, g = l m, g = k n and k = l p are reductive decompositions for M, N and F, respectively. inclusions Hence, we have the following [k, n] n, [l, n] n, [l, p] p and [p, n] n. (.3) An Ad K-invariant scalar product on n induces a G-invariant Riemannian metric g N on N and an Ad L-invariant scalar product on p induces a G-invariant Riemannian metric g F on F. The orthogonal direct sum of these scalar products on m defines a G-invariant Riemannian metric g M on M which projects onto a G-invariant metric on N. Moreover, if p and n do not contain any equivalent Ad L-submodules, then any G-invariant metric which projects onto a G-invariant metric on N is constructed in this fashion. We recall the following result due to L.Bérard-Bergery ([9], [0] 9 H): Theorem.3. The natural projection π : M N is a Riemannian submersion from (M, g M ) to (N, g N ) with totally geodesic fibers. Throughout this thesis we shall refer to such a metric g M as an adapted metric: 9

27 Definition.4. An adapted metric on M is a G-invariant Riemannian metric g M such that the natural projection π : M N is a Riemannian submersion and consequently the fibers are totally geodesic submanifolds. The fibration M N equipped with an adapted metric g is then called a Riemannian fibration. An adapted metric on M shall be denoted by g M and, as already introduced above, g F and g N shall denote the projection of g M onto the base space N and g F its restriction to the fiber F. We consider a decomposition p = p... p s into irreducible Ad L-submodules pairwise orthogonal with respect to B. Also let n = n... n n be an orthogonal decomposition into irreducible Ad K-submodules. Throughout we assume the following hypothesis: (i) p,..., p s are pairwise inequivalent irreducible Ad L-submodules; (ii) n,..., n n are pairwise inequivalent irreducible Ad K-submodules; (iii) p and n do not contain equivalent Ad L-submodules. We shall refer to this hypothesis by saying that M has simple spectrum. Under this hypothesis, according to Schur s Lemma, any Ad L-invariant scalar product on m = p n which restricts to an Ad K-invariant scalar product on n is of the form <, >= ( s a=λ a B pa pa ) ( n k=µ k B nk n k ), (.33) for some λ,..., λ s, µ,..., µ n > 0. Since an adapted metric g M on M projects onto a G-invariant Riemannian metric on N, g M is necessarily induced by a scalar product of the form (.33). To denote that g M is induced by a scalar product as in (.33) we shall write Similarly, we write g M = g M (λ,..., λ s ; µ,..., µ n ). (.34) g F = g F (λ,..., λ s ) and g N = g N ( µ,..., µ n ). (.35) By Ric we mean the Ricci curvature of g M and by Ric F and Ric N the Ricci curvature of g N and g F, respectively. In the following Sections we compute the Ricci curvature for g M and find some necessary conditions so that g M is an Einstein metric. We recall that in Theorem. we have shown that the Ricci curvature of any metric on M can be described using the bilinear maps Q XY defined in.. 0

28 .. The Ricci Curvature in the Direction of the Fiber In this Section we obtain the Ricci curvature of the adapted metric g M = g M (λ,..., λ s ; µ,..., µ n ) in the vertical direction p. We recall that p decomposes into the direct sum of the pairwise inequivalent irreducible Ad L-submodules p,..., p s and, as explained above, g M is induced by the scalar product (.33) ( s a=λ a B pa p a ) ( n k=µ k B nk n k ), while g F is the restriction of g M to the fiber, i.e., Lemma.6. Let X p and Y m. (i) Q n jp a XY = Qpan j XY g F = g F (λ,..., λ s ). = 0, a =,..., s, j =,..., n; (ii) Q n in j XY = 0, if i j, i, j =,..., n; (iii) Q n jn j XY = Kill(C n j X, Y ), j =,..., n. Proof: Since ad X p k n we have Q n jp a XY.4 (ii), Q pan j XY = Qn jp a XY = 0. As ad X n j n j,we have Q n jn j every i j, we also conclude that Q n in j XY = 0, from Lemma.5. From Lemma XY = Kill(C n j X, Y ). Moreover, since n j n i, for = 0, if i j. Lemma.7. The Ricci curvature of g F = g F (λ,..., λ s ) is of the form Ric F = s a=q a B pa p a, with where the constants q cb a q a = s b,c= and γ a are such that ( λ a λ ) c qa cb + γ a λ c λ b λ b and In particular, Ric F (p a, p b ) = 0, if a b. Kill k pa p a = γ a Kill pa p a Q p bp c pa p a = q cb a Kill pa p a. Proof: Since p,..., p s are pairwise inequivalent irreducible Ad L-submodules and Ric F is an Ad L-invariant symmetric bilinear form, we may write Ric F =

29 s a=q a B pa p a, for some constants q,..., q s. In particular, we have Ric F (p a, p b ) = 0, for every a, b =,..., s such that a b. By Theorem., the Ricci curvature of g F is Ric F (X, X) = s b,c= ( λb λ c λ a λ c λ b ) Q pcp b XX Kill k(x, X). By Lemma.4 the maps Q pcp b are Ad L-invariant symmetric bilinear maps. Since p a is Ad L-irreducible, there are constants q cb a such that Q pcp b pa p a = q cb a Kill pa p a. Similarly, the Killing form of k, Kill k, is an Ad L-invariant symmetric bilinear form on p a. So, by irreducibility of p a, there is a constant γ a such that Kill(C k, ) pa p a = γ a Kill pa p a. By the expression above for Ric F, we must have q a = s b,c= and the result follows from this. ( λ a λ ) c qa cb + γ a λ c λ b λ b Proposition.. Let g M = g M (λ,..., λ s ; µ,..., µ n ) be an adapted metric on M. For every a, b =,..., s such that a b, Ric(p a, p b ) = 0. For every X p a, a =,..., s, Ric(X, X) = ( q a + λ a 4 n j= c nj,a µ j where, for j =,..., n, the constants c nj,a are such that ) B(X, X), and C nj such that with q cb a Kill(C nj, ) pa p a = c nj,akill pa p a is the Casimir operator of n j with respect to Kill. The constant q a is and γ a defined by q a = s b,c= ( λ a λ ) c qa cb + γ a λ c λ b λ b,

30 Kill k pa p a = γ a Kill pa p a Q p bp c pa p a = q cb a Kill pa p a. Proof: Since p,..., p s are pairwise inequivalent irreducible Ad L-submodules and Ric p p is an Ad L-invariant symmetric bilinear form, we have that Ric p p is diagonal, i.e., Ric p p = a B p p... a s B ps p s, for some constants a,..., a s. In particular, we have Ric(p a, p b ) = 0, for every a, b =,..., s such that a b. Hence, Ric p p is determined by elements Ric(X, X) with X p a, a =,..., s. By Lemma.6 we obtain that only Q n jn j XX = Kill(C n j X, X) and Q p bp c XX are non-zero. Therefore, by Theorem. we obtain that s j,k= ( λk λ j λ a λ j λ k ) Q p jp k XX + Ric(X, X) = n j= ( ) λ a Q n jn j µ XX Kill(X, X). j We have m j= Qn jn j XX = m j= Kill(C n j X, Y ) = Kill(C n X, X) Hence we can rewrite Ric(X, X) as follows: = Kill(X, X) Kill(C k X, X) = Kill(X, X) Kill k (X, X). s ( ) λb λ a Q pcp b XX λ c λ c λ b Kill k(x, X) a,b= }{{} () n λ a µ j= j Kill(C nj X, X). As we saw in the proof of Lemma.7, the summand () is just Ric F (X, X) = q a B(X, X). Furthermore, since Kill(C nj, ) = Q n jn j are Ad L-invariant symmetric bilinear maps (Lemma.4) and p a is Ad L-irreducible, there are constants c nj,a such that Kill(C nj, ) pa p a = c nj,akill pa p a. 3

31 Therefore, Ric(X, X) = ( q a + 4 n λ a c µ nj,a j= j ) B(X, X)...3 The Ricci Curvature in the Horizontal Direction We obtain the Ricci curvature of an adapted metric g M = g M (λ,..., λ s ; µ,..., µ n ) in the horizontal direction n. We recall that n decomposes into the direct sum of the pairwise inequivalent irreducible Ad K-submodules n,..., n n and, as explained above, g M is induced by the scalar product (.33) ( s a=λ a B pa p a ) ( n k=µ k B nk n k ) and g N is the projection of g M onto the base space, i.e., Lemma.8. Let X n k and Y m. (i) Q n jp a XY (ii) Q pan k XY (iii) Q p jp a XY = Qpan j XY = Qn kp a = 0, for j k; XY = Kill(C p a X, Y ); = 0, for i, j =,..., s. g N = g N (µ,..., µ n ). Proof: We have ad X p a n k p, n j, for j k. Thus, Q n jp a XY = 0, for j k and Q p jp a XY = 0, for i, j =,..., s, from Lemma.5. Also from ad X p a n k we deduce that Q pan k XY = Kill(C pa X, Y ). From Lemma.4 (ii), we also obtain Q pan j XY = Qn jp a XY = 0, for j k and Qn kp a XY = Qpan k XY = Kill(C p a X, Y ), for j = k. Lemma.9. The Ricci curvature of g N = g N (µ,..., µ n ) is of the form Ric N = n k= r kb nk n k, where and the constants r ji k r k = n j,i= are such that ( µ a µ ) i r ji k µ i µ j µ + j Q n jn i nk n k = r ji k Kill n k n k. In particular, Ric N (n k, n j ) = 0, if k j. 4

32 Proof: The metric g N is induced by an Ad K-invariant scalar product on n. Hence, Ric N is an Ad K-invariant symmetric bilinear form on n. Since the subspaces n j, j =,..., n, are irreducible pairwise inequivalent Ad K-submodules, we may write Ric N = m k=r k B nk n k, for some constants r,..., r n. It is clear that Ric N (n k, n j ) = 0, if k j. It follows from Theorem. that, for every X n k, Ric N (X, X) = n j,i= ( ) µi µ k Q n jn i XX µ j µ i µ Kill(X, X). j Since each subspace n j is Ad K-invariant, the bilinear maps Q n jn i XY are Ad K- invariant symmetric bilinear maps (Lemma.4). Hence, by irreducibility of the n k s it follows that Q n jn i nk n k = r ji k Kill n k n k, for some constants r ji k, j, i, k =,..., n. By definition of the r k s it must be r k = n j,i= ( µ k µ ) i ra ji + µ i µ j µ j. Proposition.. Let g M = g M (λ,..., λ s ; µ,..., µ n ) be an adapted metric on M. For every k, j =,..., n, such that j k, Ric(n k, n j ) = 0. For every X n k, Ric(X, X) = µ k where, for every a =,..., s, C pa Kill, and the constants r ji k r k = n j,i= are such that s λ a B(C pa X, X) + r k B(X, X), a= is the Casimir operator of p a with respect to ( µ a µ ) i r ji k µ i µ j µ + j Q n jn i nk n k = r ji k Kill n k n k. Proof: Let X n k and Y n k. By Lemma.8 we have Q pap b XY a, b =,..., s. Also, Q n jp a XY = Qpan j XY Theorem. and Lemma.9 that, if k k, then = 0, for every = 0, if j k, k. Therefore, it follows from 5

33 Ric(X, Y ) = n j,i= ( µi µ ) kµ k Q n jn i XY µ j µ i µ j Kill(X, Y ) = RicN (X, Y ) = 0. Hence, Ric n n is determined by elements Ric(X, X) with X n k, k =,..., n. For X n k, by Theorem., we get Ric(X, X) = s k= ( µk λ a µ k µ k λ a ) Q pan k XX + s a= ( ) λa µ k Q n kp a XX µ k µ k λ + RicN (X, X). a From Lemma.8, we know that Q n kp a XX simplify the expression above obtaining = Qpan k XX = Kill(C p a X, X). Hence, we Ric(X, X) = s a= λ a µ k Kill(C pa X, X) + Ric N (X, X). Finally, since Ric N (X, X) = r k B(X, X), using the notation of Lemma.9, we conclude that Ric(X, X) = s a= λ a µ k B(C pa X, X) + r k B(X, X)...4 The Ricci Curvature in the Mixed Direction In the previous two sections we determined the Ricci curvature of g M = g M (λ,..., λ s ; µ,..., µ n ) on the directions of p and n. Here we obtain the Ricci curvature in the direction of p n. Proposition.3. Let g M = g M (λ,..., λ s ; µ,..., µ n ) be an adapted metric on M. For every X p a and Y n k, Ric(X, Y ) = λ aµ k 4 n j= B(C nj X, Y ), µ j where, for every j =,..., n, C nj Kill. is the Casimir operator of n j with respect to 6

34 Proof: For X p we know from Lemma.6 that Q n jp a XY = Q pan j XY = 0, for every a =,..., s, j =,..., n, and Q n in j XY = 0, if i j, i, j =,..., n, whereas Q n jn j XY = Kill(C nj X, Y ), j =,..., n. Moreover, for Y n k, since ad X ad Y p n k p, from Lemma.6 we also obtain that Q p bp c XY b, c =,..., s. Therefore, only Q n jn j XY = 0, for every = Kill(C n j X, Y ), j =,..., n, may not be zero. Furthermore, Kill(X, Y ) = 0. Hence, from Theorem. we get Ric(X, Y ) = n j= ( λ ) aµ k Q n jn j µ XY = j n j= ( λ ) aµ k Kill(C µ nj X, Y ). j On the other hand, n Kill(C nj X, Y ) = Kill(C n X, Y ) = Kill(X, Y ) Kill(C k X, Y ) = 0, j= since C k p k n. Therefore, Ric(X, Y ) = λ aµ k 4 n j= Kill(C nj X, Y ). µ j..5 Necessary Conditions for the Existence of an Adapted Einstein Metric From the expressions obtained previously for the Ricci curvature in the horizontal direction and in the direction of p n we obtain two necessary conditions for the existence of an adapted Einstein metric on M. These are restrictions on Casimir operators and shall be extremely useful in the chapters ahead. Corollary.4. Let g M = g M (λ,..., λ s ; µ,..., µ n ) be an adapted metric on M. If g M is Einstein, then the operator s a= λ ac pa nk is scalar. Proof: Let g M be an adapted metric as defined in (.33). If g M is Einstein with Einstein constant E, then, Ric n n = E <, > n n and thus Ric n n is Ad K- invariant. Therefore, by Proposition., we conclude that s a= λ ac pa n has to be Ad K-invariant. Hence, s a= λ ac pa nk is scalar, by irreducibility of n k. Corollary.5. Let g M = g M (λ,..., λ s ; µ,..., µ n ) be an adapted metric on M. The orthogonality condition Ric(p, n) = 0 holds if and only if n µ j= j Moreover, if g M is Einstein, then (.36) holds. C nj (p) k. (.36) 7

35 Proof: Let g M be an adapted metric of the form g M (λ,..., λ s ; µ,..., µ n ). From Proposition.3, we obtain that Ric(p, n) = 0 if and only if, for every X p a and Y n b, Kill ( n j= C nj µ j X, Y ) = 0. This holds if only if n C nj j= X k, for every X p. µ j If g M is Einstein with Einstein constant E, then Ric(p, n) = E < p, n >= 0. This shows the last assertion of the Corollary. The two previous Corollaries may be restated in the following way, which emphasizes the fact that the two necessary conditions obtained for existence of an Einstein adapted metric are just algebraic conditions on the Casimir operators of the submodules p a and n k. Corollary.6. If there exists on M an Einstein adapted metric, then there are positive constants λ,..., λ s such that the operator s a= λ ac pa nk is scalar. Furthermore, if g N is not the standard metric, then there are positive constants ν,..., ν n, not all equal, such that n ν j C nj (p) k. j= Proof: The assertions follow from Corollaries.4 and.5. In.5 we set ν k = /µ k. Hence, ν =... = ν n occurs when g N is the standard metric. Moreover, if ν =... = ν n, the inclusion in.5 is equivalent to C n (p) k, which always holds since C n = Id C k and C k maps p into k. So we obtain a condition on the C nj s only when g N is not standard. 8

36 CHAPTER As in Chapter, we consider a homogeneous fibration F M N, for M = G/L, N = G/K and F = K/L, where G is a compact connected semisimple Lie group and L K G are connected closed non-trivial subgroups, and an adapted metric g M on M. We consider some particular cases by imposing restrictions on the metric g M, which shall lead to simpler expressions of the Ricci curvature and thus allow us to determine further conditions for the existence of an Einstein adapted metric. Unless otherwise stated we follow the notation used in Chapter. Thus, as before, p,..., p s are the irreducible pairwise inequivalent Ad L-submodules of p, the tangent space to the fiber, and n,..., n n are the irreducible pairwise inequivalent Ad K-submodules of n, the tangent space to the base. An adapted metric g M on M is written as g M = g M (λ,..., λ s ; µ,..., µ n ) meaning that g M is induced by the scalar product ( s a=λ a B pa pa ) ( n k=µ k B nk n k ), on the tangent space m = p n of M. We assume that M has simple spectrum as in Section.... Riemannian Fibrations with Normal Fiber In this section we consider an adapted metric g M whose restriction to the fiber, g F, is a multiple of the Killing form of g. Hence, we have and g M = g M (λ,..., λ; µ }{{},..., µ n ) (.) s g F = g F (λ,..., λ) (.) }{{} s by setting λ =... = λ s = λ in (.33) and (.34). Clearly, when equipped with g F, F becomes a normal Riemannian manifold. In particular, if the Killing form 9

37 of k is a multiple of the Killing form of g, then F is a standard Riemannian manifold. This shall be the case when, for instance, p is irreducible, but it will not be the case in general. Proposition.. Let g M be an adapted metric on M of the form The Ricci curvature of g M is as follows: (i) For every X p a, Ric(X, X) = g M (λ,..., λ; µ }{{},..., µ n ). s ( q a + λ 4 n j= c nj,a µ j ) B(X, X), with q a = ( c l,a + γ ) a, where c l,a is the eigenvalue of the Casimir operator of l on p a, γ a is defined by and c nj,a is given by Kill k pa p a = γ a Kill pa p a where C nj Kill(C nj, ) pa p a = c nj,akill pa p a, is the Casimir operator of n j with respect to Kill. (ii) For every X n k, Ric(X, X) = λ µ k B(C p X, X) + r k B(X, X), where r k is as defined in Lemma.9; (iii) For every X p a and Y n k, Ric(X, Y ) = λµ k 4 n j= B(C nj X, Y ) ; µ j (iv) Ric(p a, p b ) = 0, for every a, b =,..., s such that a b, and Ric(n i, n j ) = 0, for every i, j =,..., n such that i j. Proof: (i) By Corollary., if g F is a multiple of B, then we obtain that Ric F (X, X) = ( ) + c l,a Kill k (X, X), 30

38 for X p a, where c l,a is the eigenvalue of the Casimir operator of l with respect to the Killing form of k on p a. Clearly, By recalling that c l,a Kill k(x, X) = Kill k (C l X, X) = tr(ad X p a ) Kill k pa pa = γ a Kill pa pa, we write Ric F = notation in Lemma.7, we have = Kill(C l X, X) = c l,a Kill(X, X). q a = ( γa ) + c l,a. The result then follows from this and Proposition.. ( γa + c ) l,a B and by following the (ii) follows directly from Proposition., by observing that s a= C p a = C p. (iii) The Ricci curvature in the direction p n essentially remains unchanged; the expression given is just that of Proposition.3 after replacing λ,..., λ s by λ. (iv) These orthogonality conditions are satisfied by any adapted metric on M and were shown to hold in Propositions. and.. Proposition.. Let g M (λ,..., λ; µ }{{},..., µ n ) be any adapted metric on M and s suppose that p,..., p s pairwise commute, i.e., [p a, p b ] = 0 if a b. For every X p a, Ric(X, X) = ( q a + λ a 4 n j= c nj,a µ j where all the constants are as in Proposition.. ) B(X, X), Proof: This follows from the fact that, in this case, Corollary.3 gives also Ric F (X, X) = ( ) + c l,a Kill k (X, X), as in the previous proof. Hence, the expression for Ric(X, X) is exactly the same obtained above for (i) in Proposition.. Corollary.. If there exists on M an Einstein adapted metric of the form g M (λ,..., λ; µ }{{},..., µ n ), then C p and C l are scalar on n k, k =,..., n. s 3

39 Proof: Since s a= C p a = C p, the necessary condition for g M to be Einstein given in Corollary.4, translates into the condition that C p is scalar on n j, if λ =... = λ s = λ. We have that C k = C p + C l is scalar on n j, since n j is irreducible as a K-module. Then C p is scalar on n j if and only if C l is.. Riemannian Fibrations with Standard Base In this section we consider an adapted metric g M whose projection onto the base space, g N, is a multiple of the Killing form of g. Hence, we have and g M = g M (λ,..., λ s ; µ,..., µ ) (.3) }{{} n g N = g N (µ,..., µ ), (.4) }{{} n by setting µ =... = µ n = µ in (.33) and (.34). In this particular case, when equipped with g N, N is a standard Riemannian manifold. Proposition.3. Let g M be an adapted metric on M of the form The Ricci curvature of g M is as follows: (i) For every X p a, g M (λ,..., λ s ; µ,..., µ ). }{{} n ( ) Ric(X, X) = q a + λ a 4µ ( γ a) B(X, X), where q a and γ a are as defined in Lemma.7, i.e., they are defined by the identities (ii) For every X n k, Kill k pa p a = γ a Kill pa p a and Ric F pa p a = q a B pa p a ; Ric(X, X) = s a= λ a µ B(C p a X, X) + r k B(X, X), with r k = ( ) + c k,k, where c k,k is the eigenvalue of the Casimir operator C k on n k ; (iii) Ric(p, n) = 0; (iv) Ric(p a, p b ) = 0, for every a, b =,..., s such that a b, and Ric(n i, n j ) = 0, for every i, j =,..., n such that i j. 3

40 Proof: (i) From the fact that γ a + n j= c n j,a =, we obtain n j= λ a µ c n j,a = λ a µ ( γ a). The required expression follows immediately from Proposition.. ( (ii) From Corollary. we obtain that r k = + c ) k,k, where rk is as defined in Lemma.9. The expression then follows from Proposition.. (iii) By using the fact that C n = n j= C n j, from Proposition.3 it follows that Ric(X, Y ) = λ a 4µ B(C nx, Y ), for every X p a and Y n k. Moreover, since C n = C g C k = Id C k and C k (p) k, we have that C n (X) k is orthogonal to Y n with respect to B. Hence, Ric(X, Y ) = 0. (iv) these orthogonality conditions are simply those in Propositions. and.. Corollary.. Let g M be any adapted metric on M and suppose that n,..., n n pairwise commute, i.e., [n j, n k ] = 0, for k j. Then, for every X n k, Ric(X, X) = s a= λ a µ k B(C pa X, X) + r k B(X, X), where r k = n k. ( + c ) k,k and ck,k is the eigenvalue of the Casimir operator C k on Proof: The proof is immediate by using Corollary.3 and Proposition...3 Binormal Riemannian Fibrations A G-invariant metric g M on M of the form g M = (λ,..., λ; µ,..., µ ) (.5) }{{}}{{} s n is called binormal. That is, a binormal metric is induced by the scalar product λb p p µb n n on m. The fibration F M N is then called a binormal Riemannian fibration. Clearly, a binormal metric projects onto an invariant metric on the 33