ON THE GEOMETRY OF CONFORMAL FOLIATIONS WITH MINIMAL LEAVES

Transcription

1 ON THE GEOMETRY OF CONFORMAL FOLIATIONS WITH MINIMAL LEAVES JOE OLIVER Master s thesis 015:E39 Faculty of Science Centre for Mathematical Sciences Mathematics CENTRUM SCIENTIARUM MATHEMATICARUM

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3 Abstract In this Master s thesis we will study the geometry of conformal foliations of 4-dimensional Riemannian manifolds with minimal leaves. We pay close attention to the adapted almost Hermitian structures on these 4-manifolds. In Chapter 1 we study these foliations when the 4-manifold is also Einstein and show what consequences this has on the almost Hermitian structures. Specifically we will provide an alternative proof to the result of John C Wood in [7]. In Chapter we follow [3] by Sigmundur Gudmundsson in which we will discuss counterexamples that show that the converse of Wood s result in [7] is not true in general. For this we construct a counterexample closely related to one given in [3] by using 4-dimensional Lie groups with leftinvariant conformal foliation. These Lie groups were classified in [4] by Gudmundsson and Martin Svensson. In Chapter 3 we look at these foliations (that are not necessarily Einstein) with additional geometrical constraints and see what consequences this has on the almost Hermitian structures. Specifically we will discuss a new result concerning the cosymplicity of the almost Hermitian structures, in which we will follow [] by Gudmundsson. Throughout this work it has been my firm intention to give reference to the stated results and credit to the work of others. The only results I claim are mine have been marked with an asterix [*]. All theorems, propositions, lemmas and examples left unmarked are assumed to be too well known for a reference to be given.

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5 Acknowledgments I would like to thank my supervisor Sigmundur Gudmundsson for suggesting this interesting topic and dedicating much of his time to help me understand the details. A special thanks goes to my family as without their continued support and encouragement this would not be possible. Joe Oliver

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7 Contents 1 Integrability of Almost Hermitian Structures on Einstein 4-Manifolds Minimality and Conformality Einstein Manifolds The Adapted Almost Hermitian Structures The Proof of Theorem Examples Integrable Almost Hermitian Structures on Non-Einstein Manifolds 19.1 The Structure J 1 is Integrable The Structures J 1 and J are Both Integrable Cosymplectic Almost Hermitian 4-Manifolds Cosymplectic Almost Hermitian Structures The Proof of Theorem Bibliography 31

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9 Chapter 1 Integrability of Almost Hermitian Structures on Einstein 4-Manifolds In this chapter we will provide an alternative proof to an important result of J. C. Wood in his 199 paper [7]. The proof in Wood s paper relies heavily on the earlier work [6] of J. Simons from Here we provide a proof without the aid of Simons. We prove the following statement. Theorem 1.1. [7] Let (M 4, g) be a 4-dimensional orientable Einstein manifold. Let F be a conformal foliation on M of codimension with minimal leaves. Then one of the adapted almost Hermitian structures J 1 or J is integrable. The strategy for proving Theorem 1.1, as outlined in [1] by P. Baird and Wood, would be to prove superminimality of the leaves of the foliation with respect to one of the almost Hermitian structures J 1 or J. Then as described in [7] and [1] the superminimality of the leaves of the foliation with respect to J 1 and J is directly linked to the integrability of J 1 and J. In [7] and [1] the method of proving superminimality is done by an application of the results of J. Simons from [6], namely the Jacobi equation. In the absence of the Jacobi equation from [6] we will use a series of lemmas to give useful results, namely Proposition 1.1 and 1.15 that we will use to prove superminimality with respect to J 1 or J, and therefore integrability of J 1 or J. First we will introduce some basic definitions that will be used and discussed throughout the subsequent sections. Definition 1.. [5] Let M be an n-dimensional manifold and let F = {F α } denote a partition of M into disjoint pathwise-connected subsets. Then F is called a f oliation of M of codimension c where 0 < c < n if there exists a cover of M by open sets U each equipped with a homeomorphism h : U R n which throws each non-empty component F α U onto a parallel translation of the standard subspace R n c in R n. Each F α is known as a leaf of the foliation F. Definition 1.3. Let M be a real differential manifold. An almost complex structure J on M is a tensor field J : C (T M) C (T M), such that J x = I, for all x M, where I is the identity mapping of C (T M) Also an almost Hermitian structure is a pair (J, g) consisting of an almost-complex structure J on M and a Riemannian metric g that is invariant under J, that is g(jx, JY ) = g(x, Y ) for all vector fields X, Y C (T M). In this text when referring to an almost Hermitian structure J we will assume the manifold M is equipped with a Riemannian metric g that is invariant under J. For an almost complex structure J on M, the pair (M, J) is called an almost complex manifold. Also if M is equipped with an almost Hermitian structure (J, g) then it is known as an almost Hermitian manifold. 1

10 In Definition 1.1 we introduce the notion of integrability of the almost complex (respectively, Hermitian) structure J. It is known that for an almost complex (respectively, Hermitian) manifold (M, J) that has an integrable almost complex (respectively, Hermitian) structure J, then the almost complex (respectively, Hermitian) structure is a complex (respectively, Hermitian) structure and the manifold M equipped with J is a complex (respectively, Hermitian) manifold. Definition 1.4. Let M be a topological manifold with atlas {(U α, z α ) α I}, where I is some index set, and z α (U α ) = V α is an open set in C n. M is a complex manifold if z α (z β ) 1 : z β (U α U β ) C n, are holomorphic maps, for all α, β I. The collection of charts (z α, U α ) is called a holomorphic structure. A Hermitian manifold is a complex manifold equipped with a Hermitian metric on its holomorphic tangent space. 1.1 Minimality and Conformality Let (M, g) be a Riemannian manifold, V be an involute distribution on M and denote by H its orthogonal complement distribution on M. Let V and H also denote the orthogonal projections onto the corresponding subbundles of T M and denote by F the foliation corresponding to V. The second fundamental form for V is given by B V (Z, W ) = 1 H( Z W + W Z) (Z, W V), and similarly the second fundamental form of H is given by B H (X, Y ) = 1 V( Y + X) (X, Y H). Definition 1.5. The foliation F tangent to V is said to be conformal if there is a vector field V V such that B H = g V, and F is said to be Riemannian if V = 0. Also, the foliation F is said to be minimal if and totally geodesic if trace B V = 0, B V = 0. From now on we assume that (M 4, g) is a 4-dimensional orientable Riemannian manifold equipped with a minimal and conformal foliation F of codimension. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Lemma 1.6. Let (M 4, g) be an oriented 4-dimensional Riemannian manifold with a conformal foliation F of codimension. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then i. X Y H, ii. Y + X H. Proof. Using conformality given in Definition 1.5 and as {X, Y, Z, W } is an orthonormal frame of T M we see that B H (X, X) B H (Y, Y ) = 0 and B H (X, Y ) = 0, therefore by the definition of the second fundamental form of H V( X Y ) = 1 V( X + X) 1 V( Y + Y ) = BH (X, X) B H (Y, Y ) = 0,

11 hence Also by the second fundamental form of H X Y H. so 1 V( Y + X) = BH (X, Y ) = 0, Y + X H. Lemma 1.7. Let (M 4, g) be an oriented 4-dimensional Riemannian manifold with a minimal foliation F of codimension. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then Z W V. Proof. Using minimality given in Definition 1.5 and as {X, Y, Z, W } is an orthonormal frame of T M we see that B V (Z, Z) + B V (W, W ) = 0, therefore by the definition of the second fundamental form of V H( Z W ) = 1 H( Z Z + Z Z) + 1 H( W W + W W ) = BV (Z, Z) + B V (W, W ) = 0, hence Z W V. 1. Einstein Manifolds Definition 1.8. Let (M, g) be a Riemannian manifold, then the Ricci curvature tensor, Ric : C (T M) C 0 (T M) is given by m Ric(E, F ) = R(E, e i )e i, F, i=1 E, F C (T M), where {e 1,..., e m } is any local orthonormal frame for the tangent bundle T M. Definition 1.9. A Riemannian manifold (M, g) is said to be Einstein or an Einstein manifold if the Ricci curvature tensor is proportional to the metric i.e. Ric(E, F ) = λg(e, F ) for all E, F C (T M), λ R, this is also known as the Einstein condition. The idea is to use Lemmas 1.6 and 1.7 above for conformality and minimality to get two equations induced by the Einstein condition which can be used later to prove superminimality of the foliation F. Lemma [*] Let (M 4, g) be an oriented 4-dimensional manifold with a conformal foliation F of codimension. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then Z Z, Y + W + Z Z, X + W = 0. 3

12 Proof. Note by Lemma 1.6 that Y + X H, X Y H, which implies X, E = Y, E, Y, E = X, E, E V, respectively. Now expanding and using the above Z Z, Y + W + Z Z, X + W = Z Z, Y Z, Z Y + W W, W Y +Z Z, X Z, Z X + W W, W X = Z Y, Z Z X, Z W Y, W W X, W Z, Z Y Z, Z X W, W Y W, W X = Z, X X + Z, Y Y + Z, W W, Z Y, X X + Z Y, Z Z + Z Y, W W Z, X X + Z, Y Y + Z, W W, Z X, Y Y + Z X, Z Z + Z X, W W X + Y + W, Z Z, W Y, X X + W Y, Z Z + W Y, W W X + Y + W, Z Z, W X, Y Y + W X, Z Z + W X, W W = Z, X Z Y, X Y, W Z, Y Z X, Y Z, W Z, W W Y, X W, Z W Y, Z W X, Y W, Z W X, Z = X, Z Z X, Y + + Y, Z Z X, Y + X, W W X, Y + Y, W W X, Y = X, Z Z X, Y + X, Z Z X, Y + + X, W W X, Y + X, W W X, Y = 0. Lemma [*] Let (M 4, g) be an oriented 4-dimensional manifold with a conformal foliation F of codimension with minimal leaves. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then [X, Z] Z, Y + [X, W ] + [Y, Z] Z, X + [Y, W ] = 4 Z Z + 4 Z Z, X Z Z, Y. Proof. Consider the first term [X, Z] Z, Y = X Z Z X Z, Y = ( X Z, X X + Z, Y Y + Z, W W Z X, Y Y Z X, Z Z Z X, W W ) Z, Y = Z, X Z, Y + Z, Y Z, Y + Z, W W Z, Y Z X, Y Z, Y Z X, Z Z Z, Y Z X, W W Z, Y = X, Z Y, Z + Y, Z Y, Z + + Z X, Y Y, Z + Z Z, X Z Z, Y + Z Z = X, Z Y, Z + + Z X, Y Y, Z + Z Z, X Z Z, Y + Z Z. Consider the second term [X, W ] 4

13 = X W W X = ( X X + Y + W, Z Z W X, Y Y W X, Z Z W X, W W ) = + + W, Z Z W X, Y W X, Z Z W X, W W = X, W Y, W + Y, W Y, W + W X, Y Y, W + Z Z + W W = X, W Y, W + W X, Y Y, W + Z Z + W W. Consider the third term [Y, Z] Z, X = Y Z Z Y Z, X = ( Y Z, X X + Z, Y Y + Z, W W Z Y, X X Z Y, Z Z Z Y, W W ) Z, X = Z, X Z, X + Z, Y Z, X + Z, W W Z, X Z Y, X Z, X Z Y, Z Z Z, X Z Y, W W Z, X = X, Z X, Z + Y, Z X, Z + Z X, Y X, Z + Z Z, Y Z Z, X + Z Z = Y, Z X, Z + Z X, Y X, Z + Z Z, Y Z Z, X + Z Z. Consider the fourth term [Y, W ] = Y W W Y = ( Y X + Y + W, Z Z W Y, X X W Y, Z Z W Y, W W ) = + + W, Z Z W Y, X W Y, Z Z W Y, W W = X, W X, W + Y, W X, W W X, Y X, W + Z Z + W W = Y, W X, W W X, Y X, W + Z Z + W W. Therefore [X, Z] Z, Y + [X, W ] + [Y, Z] Z, X + [Y, W ] = X, Z Y, Z + + Z X, Y Y, Z + Z Z, X Z Z, Y + Z Z + X, W Y, W + W X, Y Y, W + Z Z + W W Y, Z X, Z + Z X, Y X, Z + Z Z, Y Z Z, X + Z Z Y, W X, W W X, Y X, W + Z Z + W W 5

14 = 4 Z Z + Z Z, X Z Z, Y + W W = 4 Z Z + 4 Z Z, X Z Z, Y. Now using the Einstein condition from Definition 1.9 and Lemmas 1.10 and 1.11 we have the following result. Proposition 1.1. [*] Let (M 4, g) be an oriented 4-dimensional Einstein manifold with a conformal foliation F of codimension with minimal leaves. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then Z Z + Z Z, X Z Z, Y = 0. Proof. Consider the Einstein condition from Definition 1.9. As we have an orthonormal basis {X, Y, Z, W }, then for all E, F {X, Y, Z, W } where E F we have Therefore 0 = λg(e, F ) + λg(f, E) = Ric(E, F ) + Ric(F, E) λ R. 0 = Ric(X, Y ) + Ric(Y, X) = R(X, X)X, Y + R(X, Y )Y, Y + R(X, Z)Z, Y + R(X, W )W, Y + R(Y, X)X, X + R(Y, Y )Y, X + R(Y, Z)Z, X + R(Y, W )W, X = R(X, Z)Z, Y + R(X, W )W, Y + R(Y, Z)Z, X + R(Y, W )W, X = Z Z Z Z [X, Z] Z, Y + W W W W [X, W ] + Z Z Z Z [Y, Z] Z, X + W W W W [Y, W ] = Z Z, Y Z Z, Y [X, Z] Z, Y + W W [X, W ] + Z Z, X Z Z, X [Y, Z] Z, X + W W [Y, W ] = ( Z W ), Y + ( Z W ), X Z Z, Y [X, Z] Z, Y W [X, W ] Z Z, X [Y, Z] Z, X W [Y, W ] = Z W, Y + X Z Z, Y [X, Z] Z, Y W [X, W ] Z Z, X [Y, Z] Z, X W [Y, W ]. Now note by Lemmas 1.6 and 1.7 that Y + X H, Z W V respectively. So 0 = Z Z, Y [X, Z] Z, Y W [X, W ] Z Z, X [Y, Z] Z, X W [Y, W ], which implies 0 = Z Z, Y + W + Z Z, X + W 6

15 [X, Z] Z, Y + [X, W ] + [Y, Z] Z, X + [Y, W ]. Then by Lemmas 1.10 and 1.11 we have and so 4 Z Z + 4 Z Z, X Z Z, Y = 0, Z Z + Z Z, X Z Z, Y = 0. Now we have one of the two equations that is induced by the Einstein condition that will be applied later to prove superminimality. For the second equation consider the next two results which are very similar to Lemmas 1.10 and Lemma [*] Let (M 4, g) be an oriented 4-dimensional manifold with a conformal foliation F of codimension. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then Proof. Note by Lemma 1.6 that which implies Z Z, Y + W Z Z, X W = 0. Y + X H, X Y H, Y, E = X, E, X, E = Y, E, E V, respectively. Now expanding and using the above Z Z, Y + W Z Z, X W = Z Z, Y Z, Z Y + W W, W Y Z Z, X + Z, Z X W + W, W X = Z Y, Z + Z X, Z W Y, W + W X, W Z, Z Y + Z, Z X W, W Y + W, W X = Z, X X + Z, Y Y + Z, W W, Z Y, X X + Z Y, Z Z + Z Y, W W + Z, X X + Z, Y Y + Z, W W, Z X, Y Y + Z X, Z Z + Z X, W W X + Y + W, Z Z, W Y, X X + W Y, Z Z + W Y, W W + X + Y + W, Z Z, W X, Y Y + W X, Z Z + W X, W W = Z, X Z Y, X Y, W + Z, Y Z X, Y + X, W W Y, X W, Z W Y, Z + W X, Y + W, Z W X, Z = + Y, Z Z X, Y + Y, Z Z X, Y = 0. + Y, W W X, Y Y, W W X, Y + Lemma [*] Let (M 4, g) be an oriented 4-dimensional manifold with a conformal foliation F of codimension with minimal leaves. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then [Y, Z] Z, Y + [Y, W ] [X, Z] Z, X [X, W ] = Z + Z Z, Y Z Z Z, X. 7

16 Proof. Consider the first term [Y, Z] Z, Y = Y Z Z Y Z, Y = ( Y Z, X X + Z, Y Y + Z, W W Z Y, X X Z Y, Z Z Z Y, W W ) Z, Y = Z, X Z, Y + Z, Y Z, Y + Z, W W Z, Y Z Y, X Z, Y Z Y, Z Z Z, Y Z Y, W W Z, Y = X, Z Y, Z + Y, Z Y, Z + Z X, Y Y, Z + Z Z, Y Z Z, Y + Z Z = Y, Z + Y, Z + Z X, Y Y, Z + Z Z, Y + Z. Consider the second term [Y, W ] = Y W W Y = ( Y X + Y + W, Z Z W Y, X X W Y, Z Z W Y, W W ) = + + W, Z Z W Y, X W Y, Z Z W Y, W W = X, W Y, W + Y, W Y, W W X, Y Y, W + Z Z + W W = Y, W + Y, W W X, Y Y, W + Z + W. Consider the third term [X, Z] Z, X = X Z Z X Z, X = ( X Z, X X + Z, Y Y + Z, W W Z X, Y Y Z X, Z Z Z X, W W ) Z, X = Z, X Z, X Z, Y Z, X Z, W W Z, X + Z X, Y Z, X + Z X, Z Z Z, X + Z X, W W Z, X = X, Z X, Z Y, Z X, Z + Z X, Y Y, Z Z Z, X Z Z, X Z Z = X, Z + Y, Z + Z X, Y Y, Z Z Z, X Z. Consider the fourth term [X, W ] = X W W X = ( X X + Y + W, Z Z W X, Y Y W X, Z Z W X, W W ) = W, Z Z 8

17 + W X, Y + W X, Z Z + W X, W W = X, W X, W Y, W X, W + + W X, Y Y, W Z Z W W = X, W + Y, W + + W X, Y Y, W Z W. Therefore [Y, Z] Z, Y + [Y, W ] [X, Z] Z, X [X, W ] = Y, Z + Y, Z + Z X, Y Y, Z + Z Z, Y + Z Y, W + Y, W W X, Y Y, W + Z + W X, Z + Y, Z + Z X, Y Y, Z Z Z, X Z X, W + Y, W + + W X, Y Y, W Z W = Z + Z Z, Y Z Z Z, X. Now by following the same procedure for Proposition 1.1, we can use the Einstein condition from Definition 1.9 and Lemmas 1.13 and 1.14 to achieve the following result. Proposition [*] Let (M 4, g) be an oriented 4-dimensional Einstein manifold with a conformal foliation F of codimension with minimal leaves. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Then Z + Z Z, Y = Z + Z Z, X. Proof. Consider the Einstein condition from Definition 1.9. {X, Y, Z, W }, then for all E, F {X, Y, Z, W } we have As we have an orthonormal basis 0 = λ λ = λg(e, E) λg(f, F ) = Ric(E, E) Ric(F, F ) λ R. Therefore 0 = Ric(X, X) Ric(Y, Y ) = R(X, X)X, X + R(X, Y )Y, X + R(X, Z)Z, X + R(X, W )W, X R(Y, X)X, Y R(Y, Y )Y, Y R(Y, Z)Z, Y R(Y, W )W, Y = R(X, Z)Z, X + R(X, W )W, X R(Y, Z)Z, Y R(Y, W )W, Y = Z Z Z Z [X, Z] Z, X + W W W W [X, W ] Z Z Z Z [Y, Z] Z, Y W W W W [Y, W ] = Z Z, X Z Z, X [X, Z] Z, X + W W [X, W ] Z Z, Y + Z Z, Y + [Y, Z] Z, Y W + W + [Y, W ] = ( Z W ), X ( Z W ), Y Z Z, X [X, Z] Z, X W [X, W ] + Z Z, Y + [Y, Z] Z, Y + W + [Y, W ] 9

18 = Z W, X Y Z Z, X [X, Z] Z, X W [X, W ] + Z Z, Y + [Y, Z] Z, Y + W + [Y, W ]. Now note by Lemmas 1.6 and 1.7 that X Y H, Z W V respectively. So 0 = Z Z, X [X, Z] Z, X W [X, W ] + Z Z, Y + [Y, Z] Z, Y + W + [Y, W ], Then by Lemmas 1.13 and 1.14 we have and so Z + Z Z, Y Z Z Z, X = 0, Z + Z Z, Y = Z + Z Z, X. 1.3 The Adapted Almost Hermitian Structures Now we have the necessary tools in the form of Propositions 1.1 and 1.15 to introduce the following. Let (M 4, g) be a 4-dimensional orientable Einstein manifold equipped with a conformal foliation F of codimension with minimal leaves. Then there exist, up to sign, exactly two almost Hermitian structures J 1 and J on M, as defined in Definition 1.3, which are adapted to the orthogonal decomposition T M = V H of the tangent bundle of M. They are determined by J 1 X = Y, J 1 Y = X, J 1 Z = W, J 1 W = Z, J X = Y, J Y = X, J Z = W, J W = Z, where {X, Y, Z, W } is any local orthonormal frame for the tangent bundle T M of M such that X, Y H and Z, W V respectively. Definition The shape operator of the foliation F is defined as S(E)F = V( F E) for all E H, F V. Definition Let (M 4, g) be a 4-dimensional orientable manifold. Then the foliation F is called superminimal in M 4 with respect to the almost Hermitian structure J i where i {1, } if S(J i E)F = J i S(E)F for all E H, F V. Now we are ready to prove a crucial result with the aid of Propositions 1.1 and This proposition together with an argument outlined in [1] will then be used to prove superminimality with respect to one of the almost Hermitian structures J 1 or J. Proposition Let (M 4, g) be an orientable 4-dimensional Einstein manifold with a conformal foliation F of codimension with minimal leaves. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. Let J i for i {1, } be the adapted almost Hermitian structure defined above. Then i. S(J i Y ), S(J i X) = 0, 10

19 ii. S(J i X), S(J i X) = S(Y ), S(Y ) = S(X), S(X) = S(J i Y ), S(J i Y ). Proof. [*] Note by Lemma 1.7 that Z W V, which implies Z Z, F = W W, F for all F H. Consider S(J i Y ), S(J i X) = S(J i Y )Z, S(J i X)Z + S(J i Y )W, S(J i X)W = Z J i Y, Z Z + Z J i Y, W W, Z J i X, Z Z + Z J i X, W W + W J i Y, Z Z + W J i Y, W W, W J i X, Z Z + W J i X, W W = Z J i Y, Z Z J i X, Z + Z J i Y, W Z J i X, W + W J i Y, Z W J i X, Z + W J i Y, W W J i X, W = Z Z, X Z Z, Y Z Z Z Z W W = Z Z, X Z Z, Y Z Z = 0 by Proposition 1.1. Now consider S(J i Y ), S(J i Y ) = S(J i Y )Z, S(J i Y )Z + S(J i Y )W, S(J i Y )W = Z J i Y, Z Z + Z J i Y, W W, Z J i Y, Z Z + Z J i Y, W W + W J i Y, Z Z + W J i Y, W W, W J i Y, Z Z + W J i Y, W W = Z J i Y, Z Z J i Y, Z + Z J i Y, W Z J i Y, W + W J i Y, Z W J i Y, Z + W J i Y, W W J i Y, W = + Z Z, X Z Z, X + Z Z + Z Z + W W = Z Z, X + Z. Which also shows that, S(X), S(X) = Z Z, X + Z. Then S(J i X), S(J i X) = S(J i X)Z, S(J i X)Z + S(J i X)W, S(J i X)W = Z J i X, Z Z + Z J i X, W W, Z J i X, Z Z + Z J i X, W W + W J i X, Z Z + W J i X, W W, W J i X, Z Z + W J i X, W W = Z J i X, Z Z J i X, Z + Z J i X, W Z J i X, W + W J i X, Z W J i X, Z + W J i X, W W J i X, W = + Z Z, Y Z Z, Y + Z Z + Z Z + W W = Z Z, Y + Z. This also shows that, S(Y ), S(Y ) = Z Z, Y + Z. Then by applying Proposition 1.15 then we have the result S(J i X), S(J i X) = S(Y ), S(Y ) = S(X), S(X) = S(J i Y ), S(J i Y ). Also by Proposition 1.18 it can be immediately seen by the symmetry of the Riemannian metric and the definition of the almost Hermitian structures that S(J i X), S(J i Y ) = 0, S(X), S(Y ) = 0 and S(Y ), S(X) = 0. 11

20 It should be noted here that the shape operator and the second fundamental form are linked by the following expression B V (F, G), E = S(E)F, G for all E H, F, G V. (1.1) We will now see that minimality from Definition 1.5 can be expressed using the shape operator. By Definition 1.5 we have, due to minimality B V (Z, Z) + B V (W, W ) = 0, which shows with the introduction of the almost Hermitian structure J i, that minimality can be expressed by B V (J i F, G) = B V (F, J i G) for all F, G V, and F G, and therefore using the shape operator and (1.1), minimality can also be expressed by S(E)J i F = J i S(E)F for all E H, F V. (1.) The next argument will be an explanation of the argument outlined in [1] in which Baird and Wood explain that easy algebra can show that a variation of Proposition 1.18 implies that S(J i E)F = ±J i S(E)F for all E H, F V. We will now prove that this is true. First consider Proposition 1.18 and then employing (1.) we have 0 = S(J i Y ), S(J i X) = S(J i Y )Z, S(J i X)Z + S(J i Y )W, S(J i X)W = ( 1) i S(J i Y )J i W, ( 1) i S(J i X)J i W + S(J i Y )W, S(J i X)W = ( 1) i+1 J i S(J i Y )W, ( 1) i+1 J i S(J i X)W + S(J i Y )W, S(J i X)W = J i S(J i Y )W, J i S(J i X)W + S(J i Y )W, S(J i X)W = S(J i Y )W, S(J i X)W = S(J i Y )W, S(Y )W, so Similarly we also have S(J i Y )W, S(Y )W = 0. (1.3) S(J i Y )Z, S(Y )Z = 0, S(J i X)W, S(X)W = 0, S(J i X)Z, S(X)Z = 0. (1.4) Considering that both H and V are dimensional submanifolds of the tangent bundle T M, then J i characterises a rotation by ± π. Also using that J i is an almost Hermitian structure, it can be seen by using (1.3) and (1.4) that S(J i Y )Z = ±λj i S(Y )Z, (1.5) S(J i Y )W = ±δj i S(Y )W, (1.6) S(J i X)Z = ±µj i S(X)Z, (1.7) S(J i X)W = ±γj i S(X)W, (1.8) for λ, µ, δ, γ R. Looking carefully at (1.6), (1.5) and using (1.) then we have S(J i Y )W = ±δj i S(Y )W = ±δ( 1) i+1 J i S(Y )J i Z = ±δ( 1) i J i S(Y )Z = ±δ( 1) i+1 S(Y )Z, 1

21 = ±δ 1 λ ( 1)i+1 J i S(J i Y )Z = δ 1 λ J is(j i Y )J i W = ±δ 1 λ J i S(J i Y )W = δ 1 λ S(J iy )W. This shows that δ = ±1 and so λ = ±δ. λ Similarly using (1.7), (1.8) and (1.) we have and so the equations become for λ, µ R. Again using (1.) with (1.9) and (1.11) then µ = ±γ, S(J i Y )Z = ±λj i S(Y )Z, (1.9) S(J i Y )W = ±λj i S(Y )W, (1.10) S(J i X)Z = ±µj i S(X)Z, (1.11) S(J i X)W = ±µj i S(X)W, (1.1) ±λj i S(Y )Z = S(J i Y )Z = S(X)Z = 1 µ J is(j i X)Z = 1 µ J is(y )Z, which implies and so With Proposition 1.18 then we see that λ = ± 1 µ, λµ = ±1. (1.13) S(Y )Z, S(Y )Z = S(Y )Z, S(Y )Z + S(Y )Z, S(Y )Z = S(Y )Z, S(Y )Z + ( 1) i S(Y )J i W, ( 1) i S(Y )J i W = S(Y )Z, S(Y )Z + J i S(Y )W, J i S(Y )W = S(Y )Z, S(Y )Z + S(Y )W, S(Y )W = S(Y ), S(Y ) = S(J i Y ), S(J i Y ) = S(J i Y )Z, S(J i Y )Z + S(J i Y )W, S(J i Y )W = S(J i Y )Z, S(J i Y )Z, and so S(Y )Z, S(Y )Z = S(J i Y )Z, S(J i Y )Z. (1.14) Now using (1.), (1.9), (1.11) and (1.14) then ±λj i S(Y )Z, S(J i Y )Z = S(J i Y )Z, S(J i Y )Z = S(Y )Z, S(Y )Z = S(Y )Z, S(J i X)Z = S(Y )Z, ±µj i S(X)Z = S(Y )Z, µj i S(J i Y )Z 13

22 = µj i S(Y )Z, S(J i Y )Z, which shows λ = ±µ. (1.15) It is easily seen that (1.15) together with (1.13) and λ R immediately shows that Which has solutions and hence λ = 1. λ = ±1, µ = ±1, which together with equations (1.9), (1.10), (1.11) and (1.1) we have the required S(J i E)F = ±J i S(E)F for all E H, F V. (1.16) Proposition Let (M 4, g) be an orientable 4-dimensional Einstein manifold with a conformal foliation F of codimension with minimal leaves, then M is superminimal with respect to J 1 or J. Proof. By Proposition 1.18, using the argument described above and outlined in [1] we have from (1.16) above S(J i E)F = ±J i S(E)F for all E H, F V. Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. First assume that M is not superminimal with respect to J 1 and using (1.) then S(J 1 Y )W = J 1 S(Y )W S(X)W = S(Y )J 1 W and and and S(X)W = S(Y )Z S(J Y )W = S(Y )J W S(J Y )W = J S(Y )W S(J 1 X)W = J 1 S(X)W S(Y )W = S(X)J 1 W S(Y )W = S(Y )Z S(J X)W = S(Y )J W S(J X)W = J S(X)W S(J 1 Y )Z = J 1 S(Y )Z S(X)Z = S(Y )J 1 Z S(X)Z = S(Y )W S(J Y )Z = S(Y )J Z S(J Y )Z = J S(Y )Z S(J 1 X)Z = J 1 S(X)Z S(Y )Z = S(X)J 1 Z S(Y )Z = S(X)W S(J X)Z = S(X)J Z S(J X)Z = J S(X)Z. So therefore not superminimal with respect to J 1 forces M to be superminimal with respect to J and similarly with M not being superminimal with respect to J forces M to be superminimal with respect to J 1. Now we have this very useful Proposition that will be used later to prove integrability of the almost Hermitian structures J 1 or J. 14

23 Proposition 1.0. Let (M 4, g) be an orientable 4-dimensional Einstein manifold with a conformal foliation F of codimension with minimal leaves. Let J i for i {1, } be an adapted almost Hermitian structure defined above, then Proof. Consider Note that F V. So for E=X then noting that F Y, Y = 0 = F X, X. Now for E=Y then H( F J i E) = H(J i F E) for all E H, F V. H( F J i E) = F J i E, X X + F J i E, Y Y. H( F J i X) = F J i X, X X + F J i X, Y Y = F Y, X X + F Y, Y Y = F Y, X J i Y + F Y, Y J i X = J i ( F X, Y Y + F Y, Y X) = J i ( F X, Y Y + F X, X X) = H(J i F X), H( F J i Y ) = F J i Y, X X + F J i Y, Y Y = F X, X X F X, Y Y = + F X, X J i Y F X, Y J i X = J i ( F X, X Y + F Y, X X) = J i ( F Y, Y Y + F Y, X X) = H(J i F Y ), again noting that F Y, Y = 0 = F X, X. Definition 1.1. An almost complex structure J i for i {1, } is said to be integrable if the Nijenhus tensor is identically zero, that is N i (E, F ) = [E, F ] + J i [E, J i F ] + J i [J i E, F ] [J i E, J i F ] = 0 for all E, F C (T M). Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. It can easily be seen by Definition 1.1 that any non vanishing component of the Nijenhus tensor satisfies the following for J 1 and the following for J J 1 N 1 (X, Z) = N 1 (X, W ) = N 1 (W, X), J 1 N 1 (X, W ) = N 1 (X, Z) = N 1 (Z, X), J 1 N 1 (Y, Z) = N 1 (Y, W ) = N 1 (W, Y ), J 1 N 1 (Y, W ) = N 1 (Y, Z) = N 1 (Z, Y ), J 1 N 1 (Z, X) = N 1 (W, X) = N 1 (X, W ), J 1 N 1 (W, X) = N 1 (Z, X) = N 1 (X, Z), J 1 N 1 (Z, Y ) = N 1 (W, Y ) = N 1 (Y, W ), J 1 N 1 (W, Y ) = N 1 (Z, Y ) = N 1 (Y, Z), J N (X, Z) = N (Y, Z) = N (X, W ), J N (Y, Z) = N (X, Z) = N (Y, W ), 15

24 J N (X, W ) = N (Y, W ) = N (X, Z), J N (Y, W ) = N (X, W ) = N (Y, Z), J N (Z, X) = N (W, X) = N (Z, Y ), J N (Z, Y ) = N (W, Y ) = N (Z, X), J N (W, X) = N (Z, X) = N (W, Y ), J N (W, Y ) = N (Z, Y ) = N (W, X). Therefore, for either J 1 or J to be integrable then it suffices to show that N 1 (E, F ) = 0 where E H and F V (1.17) or N (E, F ) = 0 where E H and F V. (1.18) 1.4 The Proof of Theorem 1.1 Now we have made the necessary preparations to prove Theorem 1.1, namely Proposition 1.19 that shows that M is superminimal with respect to J 1 or J. Proposition 1.19 together with Proposition 1.0 will now be applied to show that either (1.17) or (1.18) is true and therefore prove the integrability of J 1 or J respectively. Proof. [7] Let {X, Y, Z, W } be any local orthonormal frame of the tangent bundle T M of M such that X, Y H and Z, W V. As shown earlier it suffices to show that either (1.17) or (1.18) is true. We will see that this comes from applying Lemma 1.6 and Propositions 1.19 and 1.0. First consider when E, G H and F V N i (E, F ), G = [E, F ], G + J i [E, J i F ], G + J i [J i E, F ], G [J i E, J i F ], G = E F, G F E, G E J i F, J i G + Ji F E, J ig Ji E F, J ig + F J i E, J i G Ji E J if, G + Ji F J ie, G = E G, F F E, G + E J i G, J i F + Ji F E, J ig + Ji E J ig, F + F J i E, J i G + Ji E G, J if + Ji F J ie, G = E G + Ji E J ig, F + E J i G + Ji E G, J if F E, G + Ji F E, J ig + F J i E, J i G + Ji F J ie, G. Now applying Proposition 1.0 we get N i (E, F ), G = E G + Ji E J ig, F + E J i G + Ji E G, J if. (1.19) Noting that E, G H and F, J i F V, then using Lemma 1.6 we see that for each E, G H then E G + Ji E J ig H, E J i G + Ji E G H and so N i (E, F ), G = 0 for all E, G H, F V. Now consider when E H and F, K V N i (E, F ), K = [E, F ], K + J i [E, J i F ], K + J i [J i E, F ], K [J i E, J i F ], K = E F, K F E, K E J i F, J i K + Ji F E, J ik Ji E F, J ik + F J i E, J i K Ji E J if, K + Ji F J ie, K = Ji F J ie F E, K + Ji F E + F J ie, J i K 16

25 + E F, K E J i F, J i K Ji E F, J ik Ji E J if, K. Now noting that F, K V then it is easy to see E F, K E J i F, J i K Ji E F, J ik Ji E J if, K = 0 for all F, K V. Therefore N i (E, F ), K = Ji F J ie F E, K + Ji F E + F J ie, J i K. Considering that the leaves of the foliation F are minimal and (1.) from earlier we have N i (E, F ), K = J i F J i E F E, K + J i F E + F J i E, J i K. Finally using Proposition 1.19 and the Definition 1.17 of superminimality with respect to J 1 or J then N i (E, F ), K = J i F J i E F E, K + J i F E + F J i E, J i K = (J i ) F E + F E, K + J i F E + J i F E, J i K = F E F E, K + J i F E + J i F E, J i K = 0, for all E H and F, K V. 1.5 Examples In this section we will work through an example of a 4-dimensional Lie group with a left-invariant conformal foliation. These Lie groups were classified in [4] by S. Gudmundsson and M. Svensson. We will look at an example where we have a non-abelian vertical distribution and together with the Einstein condition from Definition 1.9 show that both J 1 and J are integrable and hence the foliation F is totally geodesic. We will also look at an example with an abelian vertical distribution and conclude that either the foliation F is totally geodesic or exactly one of the almost Hermitian structures J 1 or J is integrable. Let G be a 4-dimensional Lie group equipped with a left-invariant Riemannian metric. Let {X, Y, Z, W } be an orthonormal basis for the Lie algebra g, where as before Z, W generate a twodimensional, left-invariant and integrable distribution V on G which is conformal and has minimal leaves. Let the horizontal distribution orthogonal to V be generated by X, Y and denoted H. Then the basis {X, Y, Z, W } can be chosen so that the Lie bracket relations for g are of the form [W, Z] = λw, [Z, X] = αx + βy + z 1 Z + w 1 W, [Z, Y ] = βx + αy + z Z + w W, [W, X] = ax + by + z 3 Z z 1 W, [W, Y ] = bx + ay + z 4 Z z W, [Y, X] = rx + θ 1 Z + θ W, where λ, α, β, θ 1, θ, a, b, r, z 1, z, w 1, w R. For the examples it would be useful to state the following result describing the geometry of the situation above given by Gudmundsson in [3]. Proposition 1.. [3] Let G be a 4-dimensional Lie group and {X, Y, Z, W } be an orthonormal basis for its Lie algebra g as above. Let F be the left-invariant conformal foliation generated by g. Then 17

26 i. F is totally geodesic if and only if z 1 = z = z 3 + w 1 = z 4 + w = 0, ii. F is Riemannian if and only if α = a = 0, iii. H is integrable if and only if θ 1 = θ = 0, iv. J 1 is integrable if and only if z 1 z 4 w = z + z 3 + w 1 = 0, v. J is integrable if and only if z 1 + z 4 + w = z z 3 w 1 = 0. Example 1.3. Considering an example with a non-abelian vertical distribution, specifically the family g 3 (α, β, w 1, w, θ ) of [4], which has non-vanishing coefficients α, β, w 1, w, and θ, then we obtain the following family satisfying the Lie bracket relations [W, Z] = αw, [Z, X] = αx + βy + w 1 W, [Z, Y ] = βx + αy + w W, [Y, X] = θ W. Now assuming that the manifold in Einstein and using Propositions 1.1 and 1.15 with the Koszul formula Y, Z = [Z, X], Y + [Z, Y ], X + Z, [X, Y ], (1.0) we see that as {X, Y, Z, W } is an orthonormal basis for g then we have Therefore w 1 w = 0 and w w 1 = 0. w 1 = w = 0 and hence by Proposition 1. both J 1 and J are integrable and also F is totally geodesic. Example 1.4. Considering an example with an abelian vertical distribution, specifically the family g 9 (z, z 3, z 4, θ 1, θ ) of [4], which has non-vanishing coefficients z, z 3, z 4, θ 1 and θ then we obtain the following family satisfying the Lie bracket relations [Z, Y ] = z Z, [W, X] = z 3 Z, [W, Y ] = z 4 Z z W, [Y, X] = z X + θ 1 Z + θ W. Now assuming the manifold is Einstein and using Propositions 1.1 and 1.15 with the Koszul formula (1.0), we have that z 3 z 4 = 0 and z 4 + 4z z 3 = 0. So if z 3 = 0 then z 4 + 4z = 0, which has real solutions z 4 = z = 0 and hence by Proposition 1. both J 1 and J are both integrable and also F is totally geodesic. If z 4 = 0 then 4z z 3 = 0, which implies that z 3 = ±z and hence by Proposition 1. either J 1 or J is integrable, and additionally the foliation F is Riemannian. 18

27 Chapter Integrable Almost Hermitian Structures on Non-Einstein Manifolds In [3] by Gudmundsson from 015 an open question was answered that arose naturally from Wood s 199 paper [7]. This question is that if we have a 4-dimensional orientable manifold (M 4, g) with a conformal foliation F on M of codimension with minimal leaves, then does the integrability of the adapted almost Hermitian structure J 1 or J imply that the manifold M is Einstein? We will see a counterexample for the case that one of the adapted almost Hermitian structures is integrable, which is closely related to an example given by Gudmundsson in [3]. We will also see a counterexample for the case when both adapted Hermitian structures are integrable, and hence the foliation F totally geodesic, given by Gudmundsson in [3]. These examples will be constructed from 3-dimensional families of 4-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension, and will be special cases of the classifications given by Gudmundsson and Svensson in [4]..1 The Structure J 1 is Integrable Considering Proposition 1. then J 1 is integrable if and only if z 1 z 4 w = z + z 3 + w 1 = 0. (.1) For a 4-dimensional Lie group equipped with a left-invariant Riemannian metric and integrable Hermitian structure J 1, then from the Lie bracket relations above the basis {X, Y, Z, W } can be chosen so that the Lie bracket relations for g are of the form [W, Z] = λw, [Z, X] = αx + βy + z 1 Z (z + z 3 )W, [Z, Y ] = βx + αy + z Z + (z 1 z 4 )W, [W, X] = ax + by + z 3 Z z 1 W, [W, Y ] = bx + ay + z 4 Z z W, [Y, X] = rx + θ 1 Z + θ W, where λ, α, β, θ 1, θ, r, a, b, z 1, z R. We will now see a counterexample closely related to Gudmundsson s example given in [3], which shows that the integrability of the almost Hermitian structure J 1 does not imply that the 4-dimensional Lie group G is Einstein. 19

28 Example.1. Let M be a 4-dimensional Lie group equipped with a left-invariant Riemannian metric, an abelian vertical distribution and non-vanishing coefficients z, z 3, z 4, θ 1 and θ. Then the basis {X, Y, Z, W } can be chosen so that the Lie bracket relations for the Lie algebra g are of the form [Z, Y ] = z Z, [W, X] = z 3 Z, [W, Y ] = z 4 Z z W, [Y, X] = z X + θ 1 Z + θ W, which is the family g 9 (z, z 3, z 4, θ 1, θ ) of [4]. Then using the condition that J 1 is integrable (.1) these Lie bracket relations become [Z, Y ] = z Z, [W, X] = z Z, [W, Y ] = z W, [Y, X] = z X + θ 1 Z + θ W. It should be noted that the horizontal distribution H is not integrable and the leaves of the vertical foliation V are not totally geodesic. Using the Koszul formula (1.0) we have X = X, X X + X, Y Y + X, Z Z + X, W W = [Y, X], X Y + [Z, X], X Z + [W, X], X W = z Y, Y = Y, X X + Y, Y Y + Y, Z Z + Y, W W = [X, Y ], X X + 1 ( [Z, X], Y + [Z, Y ], X + [X, Y ], Z )Z + 1 ( [W, X], Y + [W, Y ], X + [X, Y ], W )W = z X 1 θ 1Z 1 θ W, Z = Z, X X + Z, Y Y + Z, Z Z + Z, W W = [X, Z], X X + 1 ( [Y, X], Z + [Y, Z], X + [X, Z], Y )Y + 1 ( [W, X], Z + [W, Z], X + [X, Z], W )W = 1 θ 1Y z W, W = X + Y + W, Z Z + W, W W = [X, W ], X X + 1 ( [Y, X], W + [Y, W ], X + [X, W ], Y )Y + 1 ( [Z, X], W + [Z, W ], X + [X, W ], Z )Z = 1 θ Y + z Z, X = X, X X + X, Y Y + X, Z Z + X, W W 0

29 = [Y, X], Y Y + 1 ( [Z, Y ], X + [Z, X], Y + [Y, X], Z )Z + 1 ( [W, Y ], X + [W, X], Y + [Y, X], W )W = 1 θ 1Z + 1 θ W, Y = Y, X X + Y, Y Y + Y, Z Z + Y, W W = [X, Y ], Y Y + [Z, Y ], Y Z + [W, Y ], Y W = 0, Z = Z, X X + Z, Y Y + Z, Z Z + Z, W W = 1 ( [X, Y ], Z + [X, Z], Y + [Y, Z], X )X + [Y, Z], Y Y + 1 ( [W, Y ], Z + [W, Z], Y + [Y, Z], W )W = 1 θ 1X, W = X + Y + W, Z Z + W, W W = 1 ( [X, Y ], W + [X, W ], Y + [Y, W ], X )X + [Y, W ], Y Y + 1 ( [Z, Y ], W + [Z, W ], Y + [Y, W ], Z )W = 1 θ X, Z X = Z X, X X + Z X, Y Y + Z X, Z Z + Z X, W W = 1 ( [Y, Z], X + [Y, X], Z + [Z, X], Y )Y + [Z, X], Z Z + 1 ( [W, Z], X + [W, X], Z + [Z, X], W )W = 1 θ 1Y z W, Z Y = Z Y, X X + Z Y, Y Y + Z Y, Z Z + Z Y, W W = 1 ( [X, Z], Y + [X, Y ], Z + [Z, Y ], X )X + [Z, Y ], Z Z + 1 ( [W, Z], Y + [W, Y ], Z + [Z, Y ], W )W = 1 θ 1X + z Z, Z Z = Z Z, X X + Z Z, Y Y + Z Z, Z Z + Z Z, W W = [X, Z], Z X + [Y, Z], Z Y + [W, Z], Z W = z Y, Z W = Z X + Z Y + Z W, Z Z + Z W, W W = 1 ( [X, Z], W + [X, W ], Z + [Z, W ], X )X + 1 ( [Y, Z], W + [Y, W ], Z + [Z, W ], Y )Y + [Z, W ], Z Z 1

30 = z X, W X = W X, X X + W X, Y Y + W X, Z Z + W X, W W = 1 ( [Y, W ], X + [Y, X], W + [W, X], Y )Y + 1 ( [Z, W ], X + [Z, X], W + [W, X], Z )Z + [W, X], W W = 1 θ Y z Z, W Y = W Y, X X + W Y, Y Y + W Y, Z Z + W Y, W W = 1 ( [X, W ], Y + [X, Y ], W + [W, Y ], X )X + 1 ( [Z, W ], Y + [Z, Y ], W + [W, Y ], Z )Z + [W, Y ], W W = 1 θ X z W, W Z = W Z, X X + W Z, Y Y + W Z, Z Z + W Z, W W = 1 ( [X, W ], Z + [X, Z], W + [W, Z], X )X + 1 ( [Y, W ], Z + [Y, Z], W + [W, Z], Y )Y + [W, Z], W W = z X, W W = W X + W Y + W W, Z Z + W W, W W = [X, W ], W X + [Y, W ], W Y + [Z, W ], W Z = z Y. The idea now is to use the above expressions to find equalities for the Ricci curvature and concluding that none of the family g 9 (z, z 3, z 4, θ 1, θ ) of [4] can be Einstein. Using the definition of sectional curvature R(X, Y )Y, X = Y Y [X, Y ] Y, X, (.) we get the following equality R(X, Y )Y, X = Y, X Y, X [X, Y ] Y, X and similarly for = X( Y, X ) Y, X Y ( Y, X ) + Y, X z Y, X + θ 1 Z Y, X + θ W Y, X = 1 4 θ θ 4z 1 θ 1 1 θ = 3 4 (θ 1 + θ ) 4z, R(X, Z)Z, X = 1 4 θ 1 z, R(X, W ) = 1 4 θ z, R(Y, Z)Z, Y = 1 4 θ 1 z, R(Y, W ) = 1 4 θ z, R(Z, W )W, Z = z.

31 Now considering the definition of the Ricci curvature in Definition 1.8 Ric(E, F ) = m R(E, e i )e i, F, i=1 where {e 1,..., e m } is any local orthonormal frame for the tangent bundle T M and E, F C (T M). Then we immediately have Ric(X, X) = R(X, X)X, X + R(X, Y )Y, X + R(X, Z)Z, X + R(X, W ) = 3 4 (θ 1 + θ ) 4z θ 1 z θ z = 3 4 θ θ θ θ 6z = 1 θ 1 1 θ 6z, Ric(Z, Z) = R(Z, X)X, Z + R(Z, Y )Y, Z + R(Z, Z)Z, Z + R(Z, W )W, Z = 1 4 θ 1 z θ 1 z + z = 1 θ 1. As {X, Y, Z, W } is an orthonormal basis, if we assume that the manifold is Einstein then from Definition 1.9 we have Ric(E, F ) = λ E, F for all, E, F C (T M), λ R, which implies that and so which gives Ric(X, X) = Ric(Z, Z), 1 θ 1 1 θ 6z = 1 θ 1, θ 1 1 θ 6z = 0, where, for real constants z, θ 1 and θ the only solution is z = θ 1 = θ = 0, which shows that for non-vanishing coefficients z, θ 1 and θ that none of these Riemannian Lie groups is an Einstein manifold.. The Structures J 1 and J are Both Integrable In this section we will see a counterexample for the case when both structures J 1 and J are integrable. This can be described by Proposition 1. where the structures J 1 and J are both integrable if and only if z 1 z 4 w = z + z 3 + w 1 = 0, z 1 + z 4 + w = z z 3 w 1 = 0. Again by Proposition 1. we can see that the foliation F is totally geodesic in this situation, that is z 1 = z = z 3 + w 1 = z 4 + w = 0. 3

32 Now just as the case of J 1 being integrable in Example.1 and because we have a 4-dimensional Lie group equipped with a left-invariant Riemannian metric with adapted integrable Hermitian structures J 1 and J, then the basis {X, Y, Z, W } can be chosen so that the Lie bracket relations for g are of the form [W, Z] = λw, [Z, X] = αx + βy z 3 W, [Z, Y ] = βx + αy z 4 W, [W, X] = ax + by + z 3 Z, [W, Y ] = bx + ay + z 4 Z, [Y, X] = rx + θ 1 Z + θ W, where λ, α, β, θ 1, θ, r, a, b, z 3, z 4 R. Example.. [3] Consider g 3 (α, β, w 1, w, θ ) from [4] by Gudmundsson and Svensson, which has a non-abelian vertical distribution. Now imposing that the structures J 1 and J are both integrable, from Proposition 1. we obtain a family satisfying the Lie bracket relations for nonvanishing coefficients α, β and θ [W, Z] = αw, [Z, X] = αx + βy, [Z, Y ] = βx + αy, [Y, X] = θ W, which are the special cases g 3 (α, β, 0, 0, θ ) of g 3 (α, β, w 1, w, θ ) in [4]. It should be noted that the horizontal distribution H is not integrable. Now with the use of the Koszul formula (1.0), the Levi-Civita connection satisfies X = X, X X + X, Y Y + X, Z Z + X, W W = [Y, X], X Y + [Z, X], X Z + [W, X], X W = αz, and similarly as in the previous example Y = 1 θ W, Z = αx, W = 1 θ Y, X = 1 θ W, Y = αz, Z = αy, W = 1 θ X, Z X = βy, Z Y = βx, Z Z = 0, Z W = 0, W X = 1 θ Y, W Y = 1 θ X, W Z = αw, W W = αz. Just as in the previous example the idea is to use the above expressions to find equalities for the Ricci curvature and concluding that there exists non-einstein manifolds within the family g 3 (α, β, 0, 0, θ ). Now using the definition of sectional curvature (.) given above, we have that R(X, Y )Y, X = Y, X Y, X [X, Y ] Y, X = X( Y, X ) Y, X Y ( Y, X ) + Y, X +θ W Y, X = α 3 4 θ, 4

33 and similarly for R(X, Z)Z, X = α, R(X, W ) = 1 4 θ α, R(Y, Z)Z, Y = α, R(Y, W ) = 1 4 θ α, R(Z, W )W, Z = 4α. Again considering the definition of the Ricci curvature in Definition 1.8, then we immediately have Ric(X, X) = R(X, X)X, X + R(X, Y )Y, X + R(X, Z)Z, X + R(X, W ) = α 3 4 θ α θ α = 1 θ 4α, Ric(Z, Z) = R(Z, X)X, Z + R(Z, Y )Y, Z + R(Z, Z)Z, Z + R(Z, W )W, Z = α α 4α = 6α. Now again assuming the manifold is Einstein then Ric(X, X) = Ric(Z, Z), and so 1 θ 4α = 6α, which gives θ = 4α. Showing that if 4α θ then our Riemannian Lie group is not an Einstein manifold. 5

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35 Chapter 3 Cosymplectic Almost Hermitian 4-Manifolds In this chapter we will introduce cosymplecticity of the adapted Hermitian structures and discuss a result given by Gudmundsson in [] which describes the relationship between the geometry of the foliation F and the cosymplecticity of these adapted Hermitian structures. We will prove the following theorem. Theorem 3.1. [] Let (M 4, g) be an orientable 4-dimensional Riemannian manifold. Let F be a conformal foliation on M of codimension with minimal leaves. Then the corresponding adapted almost Hermitian structures J 1 and J are both cosymplectic if and only if F is Riemannian and its horizontal distribution H is integrable. 3.1 Cosymplectic Almost Hermitian Structures Definition 3.. An almost Hermitian manifold (M, g, J) is said to be cosymplectic if its almost complex structure J is divergence f ree i.e. δj = divj = m (k J)(X k ) = 0, where {X 1,..., X m } is any local orthonormal frame for the tangent bundle T M of M. k=1 To prove Theorem 3.1 we will need the lemma below which is a specific 4-dimensional case of a proposition given by Gudmundsson in []. Lemma 3.3. Let (M 4, g) be an orientable 4-dimensional Riemannian manifold. Let F be a conformal foliation on M of codimension with minimal leaves. Then Proof. First consider HδJ i = 0 for i {1, } δj i = ( J i )X + ( J i )Y + ( Z J i )Z + ( W J i )W = J i X J i X + J i Y J i Y + Z J i Z J i Z J i W J i W W = Y J i ( X + Y + Z W ) X + ( 1) i+1 Z W + ( 1) i W Z = Y, X X + Y, Z Z + Y, W W X, Y Y X, Z Z X, W W + ( 1) i+1 Z X + ( 1) i+1 Z Y + ( 1) i+1 Z W, Z Z +( 1) i W Z, X X + ( 1) i W Z, Y Y + ( 1) i W Z, W W J i ( X, Y Y + X, Z Z + X, W W ) J i ( Y, X X + Y, Z Z + Y, W W ) 7

36 J i ( Z Z, X X + Z Z, Y Y + Z Z, W W ) J i ( W X + W Y + W W, Z Z). Note by Lemmas 1.6 and 1.7 that Y + X H, Z W V respectively. Then we have HδJ i = Y, X X X, Y Y + ( 1) i+1 Z X + ( 1) i+1 Z Y +( 1) i W Z, X X + ( 1) i W Z, Y Y J i X, Y Y J i Y, X X J i Z Z, X X J i Z Z, Y Y J i W X J i W Y = Y, X X X, Y Y J i X, Y Y J i Y, X X J i Z Z, X X J i Z Z, Y Y J i W X J i W Y = X, Y X + Y, X Y + X, Y X Y, X Y J i Z X J i Z Y = 0 3. The Proof of Theorem 3.1 In this section we will use Lemma 3.3 to prove Theorem 3.1 which was given in [] by Gudmundsson. Proof. [] Let us assume the almost complex structures J 1 and J are cosymplectic i.e for i = 1, we have 0 = δj i = ( J i )X + ( J i )Y + ( Z J i )Z + ( W J i )W = [X, Y ] + ( 1) i [W, Z] J i ( X + Y + Z W ). Then it follows from Lemma 3.3 that 0 = δj 1 + δj = VδJ 1 + VδJ = V[X, Y ] V(J 1 + J )( X + Y + Z W ) = V[X, Y ]. This shows us that the horizontal distribution H is integrable. Then employing the fact that J 1 is cosymplectic, then we see that J 1 V( X + Y ) = VJ 1 ( X + Y ) = V[W, Z] VJ 1 ( Z W ) = W Z, W W + Z W, Z Z J 1 V( Z W ) = W W, Z W Z Z, W Z J 1 ( Z Z, W W + W W, Z Z) = 0. Further it follows from V[X, Y ] = 0 and V( X + Y ) = 0 that VδJ = 0 is equivalent to V[W, Z] VJ ( Z W ) = 0. 8

37 The fact that F is conformal and using Definition 1.5 with Lemma 1.6 then B H (X, X) = B H (X, X) + B H (Y, Y ) = V( X + Y ) = 0. Since the second fundamental form B H of the horizontal distribution H is symmetric then B H 0 and so by Definition 1.5, F is Riemannian. It is easily seen from the above calculations that the other part of the statement is also valid. 9

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39 Bibliography [1] P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, London Math. Soc. Monogr. 9, Oxford Univ. Press (003). [] S. Gudmundsson, Holomorphic harmonic morphisms from cosymplectic almost Hermitian manifolds, Geometriae Dedicata, (015), DOI: /s [3] S. Gudmundsson, Holomorphic harmonic morphisms from four-dimensional non-einstein manifolds, Internat. J. Math. 6 (015), , 7 pages. [4] S. Gudmundsson, M. Svensson, Harmonic morphisms from four-dimensional Lie groups, J. Geom. Phys. 83 (014), [5] D. Rolfsen, Knots and Links, Math. Lecture Series 7, Publish or Perish Inc., Berkeley, CA (1976). [6] J. Simons, Minimal Varieties in Riemannian Manifolds, Ann. of Math. 88 (1968), [7] J. C. Wood, Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Internat. J. Math. 3 (199),

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42 Master s Theses in Mathematical Sciences 015:E39 ISSN LUNFMA Mathematics Centre for Mathematical Sciences Lund University Box 118, SE-1 00 Lund, Sweden