Invariant submanifolds in flow geometry

J. Austral. Math. Soc. (Series A) 62 (1997), 290-314

INVARIANT SUBMANIFOLDS IN FLOW GEOMETRY J. C. GONZALEZ-DA VILA, M. C. GONZALEZ- DA VILA and L. VANHECKE (Received 30 August 1995; revised 11 September 1996)

Communicated by R. Bartnik Abstract We begin a study of invariant isometric immersions into Riemannian manifolds (A/, g) equipped with a Riemannian flow generated by a unit Killing vector field f. We focus our attention on those (M, g) where f is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided. 1991 Mathematics subject classification (Amer. Math. Soc): 53B25, 53C12, 53C15, 53C25, 53C3O, 53C35, 53C40. Keywords and phrases: Isometric, normal and contact flows, Killing-transversally symmetric spaces, normal fow space forms, invariant immersions and embeddings.

0. Introduction Symmetric and Hermitian symmetric spaces play in important role in real and complex geometry. The analogs of the last class for contact geometry are the ^-symmetric spaces. They form a subclass of the Riemannian spaces equipped with a complete unit Killing vector field such that the reflections with respect to its integral curves are global isometries. These last spaces have been introduced and studied from the global and local viewpoint in [5, 6, 8] and related papers, and are called globally Killing-transversally symmetric spaces. For the different cases, the corresponding space forms, that is, real, complex, Sasakian and normalflowspace forms [7], provide classes of particularly interesting examples. Isometric immersions and embeddings in real, complex and Sasakian manifolds and their associated space forms have been studied extensively. In this paper we begin a study of isometric immersions into manifolds equipped with a Riemannian First two authors supported by the "Consejeria de Educaci6n del Gobierno de Canarias" © 1997 Australian Mathematical Society 0263-6115/97 $A2.00 + 0.00

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flow generated by a unit Killing vector field, thereby concentrating mainly on Killingtransversally symmetric spaces and normal flow space forms. In Section 1 we give some preliminary definitions, formulas and results, and in Section 2 we introduce and derive some results concerning invariant immersions of manifolds with respect to flows such that the image is tangent to the flow lines. In Section 3 we consider the case of Riemannian manifolds such that the flow gives rise to afibrationand study the relation between the total space M and the base space M', in particular with respect to congruence theorems for the immersions into M and the corresponding transversal immersions into M'. Section 4 is devoted to the case of a normal space form M. Several general results are derived and examples are provided for the two cases of normal flow space forms considered in flow geometry [7].

1. Isometric flows Let (M, g) be a n-dimensional, smooth Riemannian manifold with n > 2, which is supposed to be connected where necessary. Furthermore, let V denote the Levi-Civita connection of (M, g) and R the corresponding Riemannian curvature tensor with the sign convention Ruv = V [t/ v] — [Vj/, V v ]

for U, V e i£~(M), the Lie algebra of smooth vector fields on M. A tangentially oriented foliation of dimension one on (M, g) is called a.flow.The leaves of this foliation are the integral curves of a non-singular vector field on M and hence, after normalization, a flow is also given by a unit vector field. In particular, a non-singular Killing vector field defines a Riemannianflowand such a flow is said to be an isometricflow.See [25] for more information. In this paper we consider and denote by &$ an isometric flow generated by a unit Killing vector field £. The flow lines of ^ are geodesies and moreover, a geodesic which is orthogonal to £ at one of its points, is orthogonal to it at all of its points. Such geodesies are called transversal or horizontal geodesies. ^ determines locally a Riemannian submersion. For each m e (A/, g), let ^ be a small open neighborhood of m such that £ is regular on °M. Then the mapping n: W —»• "&" = ^/£ is a submersion. Furthermore, let g' denote the induced metric on 0 and this is impossible because the flow lines are closed submanifolds. Note that when ^ is a contact flow, then this result also follows from the wellknown Boothby-Wang theory for homogeneous contact manifolds [3]. In this context it is worthwile to mention that it follows from [6] that a KTS-space (M, g, ^ ) has a contact flow ^ if and only if it is locally irreducible. Moreover, a locally reducible KTS-space is locally isometric to a product of a contact KTS-space and a symmetric space. We also recall that when (M, g, ^ ) is a KTS-space, then (A/', g') is a symmetric space [6].

2. Invariant immersions with respect to a flow Let / be an isometric immersion of a n-dimensional Riemannian manifold (M, g) into an n-dimensional Riemannian manifold (M, g). In what follows, and if the argument is local, we shall always identify M with its image under / to simplify the notation. Next, let V and V denote the Levi-Civita connection of (M, g) and (M, g), respectively, and let R, R be the corresponding curvature tensors. The Gauss and

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Weingarten formulas are, respectively: (2.1)

VXY = VXY + B(X,Y),

(2.2)

VXU ^-CVX

+ VXU,

where X,Y & 5£{M), U € SC(M)L and V1- is the connection in the normal bundle TLM. Here B and C denote the corresponding second fundamental tensors and are related by g(B(X, Y), U) = g(CuX, Y). Next, we recall the fundamental Gauss, Codazzi and Ricci equations: (2.3) R(X, Y, Z, W) = R(X, Y, Z, W) + g(B(X, W), B(Y, Z)) -g{B(X,Z),B(Y,W)), (2.4)

(RxvZ^ = -(VXB)(Y, Z) + (VrB)(X, Z),

(2.5) R(X, Y, U, V) = RX(X, Y, U, V) - giCuX, CVY) + g(CvX, CVY) for X, Y,Z,W e SC{M), U, V e 3C{MY and where RL denotes the curvature tensor of V-1. VB is defined by (2.6)

(VXB)(Y, Z) = VX(B(Y, Z)) - B(VXY, Z) - B(Y, VXZ)

for all X, Y, Z e &(M). The immersion / is said to be totally geodesic if B = 0, parallel if VB = 0 and minimal if tr B = 0 on M. Now, let ^ be an isometric flow on (M, g) and let H be the tensor field defined in (1.1). Then / is said to be tangent if f is tangent to f(M). In this case we shall denote by £ the vector field on M given by /»§ = | o / . In what follows we will consider a special kind of tangent isometric immersions which we define now. DEFINITION 2.1. An isometric immersion / of (A/, g) into (M, g, J ^ ) is said to be invariant (with respect to theflowJ?|) if / is tangent and invariant with respect to H\ that is, Hf»TmM c f*TmM for all m e M.

For invariant immersions we have H(f*TmM)L c (ftTmM^ and the tensor field H on M related to H by / , o H = H o / , coincides with the one defined via (1.1) for the flow ^ on (M, g). Moreover, since / is tangent, we have HX = HX — B(X, §) for any X tangent to M and so we get (2.7)

B(X, £) = 0,

for all X e 3C(M) and all U e

Q/S = 0

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We note that if ^ is a contact flow on (M, g), then ^ is a contact flow too, and hence M is also odd-dimensional. The notion of an invariant immersion corresponds to a similar one in Sasakian geometry and many properties about these immersions may be proved easily in our case. We give some examples. PROPOSITION 2.1. Let ^ be a contact flow on (Af, g) and f : (Af, g) -> (Af, g) an isometric immersion. If HX is tangent to f(M)for all X tangent to M, then f is tangent and hence invariant. PROOF. First, suppose f o / is normal. Then h(X, Y) = {fj([X, Y]) = 0 for all X,Y e 3£(M} and so H cannot be of maximal rank. So, put f ° / = | r + £A' where | r and %N denote the tangential and normal part of f along M, respectively. Since H%T = H%N = 0 and since ^ is contact, we obtain that | is tangent to /(Af) along /(Af). This proves the required result.

Next, we obtain easily LEMMA 2.1. Let / : (Af, g) —>• (A?, g) be an invariant immersion and let ^ normal flow on (Af, g). Then &$ is also normal and moreover, we have (2.8) (2.9)

be a

(V x tf)y = (VXH)Y, B(X, HY) = HB(X, Y) = B(HX, Y),

(2.10)

HCVX = -CVHX

= CjjuX.

From this we then get PROPOSITION 2.2. Let ^ be a contact normal flow on (M, g) and f an invariant immersion of(M, g) into (M, g). Then f is minimal. For each m € Af we can choose an orthonormal basis {M2,_I, u2i; | ; v2J-i, v2j,; i = 1 , . . . , (n - l)/2, y = 1 , . . . , ( « - n - l)/2} of TmM, where («, «„, | ) span TmAf and (v\,... , u^-n_i) span T^M, and positive numbers A.,-, /Xj such that PROOF.

Hu2i = — Hv2J = Then tr Bm = 0 follows at once by using (2.7) and (2.9). Next, we focus on the sectional curvatures and prove

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PROPOSITION 2.3. Let f: (M, g) —»• (M, g, ^)) be an invariant immersion. Then for each X € 3E{M) and orthogonal to £ we have (ii) if &-% is normal, then K(X, HX) < K(X, HX). Moreover, if J ^ is also contact, then equality holds if and only if f is totally geodesic. PROOF, (i) follows immediately from (2.3) and (2.7). To prove (ii) one just uses the following consequence of (2.3) and (2.9): R(X, HX, X, HX) = R(X, HX, X, HX) + 2g(HB{X, X), HB(X, X)). Furthermore, we have PROPOSITION 2.4. Let ^ be a normal contact flow on (M, g) and f an invariant immersion. If f is parallel, then it is totally geodesic. PROOF.

From (2.6) and (2.9) we obtain 0 = (VXB)(Y, f) = B(Y, HX) = HB(X, Y).

Now the result follows at once since H has maximal rank. This result shows that the notion of a parallel invariant immersion is very restrictive. Therefore, we introduce the notion of an rj-parallel immersion. Let J^i be an isometric flow on (M, g) and let / be a tangent isometric immersion of (M, g) into (M, g). Then / is said to be r]-parallel if (VXB)(Y, Z) = 0 for all X, Y, Z e %(M) and orthogonal to f. Then we have PROPOSITION 2.5. Let f be an rj-parallel invariant immersion of (M, g) into (M, g, &\). If(M, g, &i) is a lo-cally KTS-space, then (M, g, &$) is also a locally KTS-space. PROOF.

From (2.3), (2.4) and (2.6) we get

(yxR)(X, Y, X, Y) =(VXR)(X, Y, X, Y) - g((VxB)(X, X), B{Y, Y)) + g(4(VxB)(X,

Y) - 2(VYB)(X, X), B(X, Y))

+ g(2(VYB)(X, Y) - 3(VXB)(Y, Y), B(X, X)). Now the result follows from this relation by using Proposition 1.2 and Lemma 2.1.

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3. Fibrations and congruent immersions Let / be a tangent isometric immersion of (M, g) into (M, g,&\) and suppose that «^| and the induced flow ^ on M are fibrable. We denote by (M, M', n) and (M, M', n) the corresponding fibre bundles. Then there exists an isometric immersion / ' : (A/', g') -+ (Af\ g') satisfying f on = n o f. Specifically, f'(m') = nf(m) with m' = 7r(m), m' e M'. This / ' is called the tranverse mapping of / . Note that when / is one-to-one, then also / ' is one-to-one. Moreover, since n is an open map [22, Corallary 3], it follows that if / is an embedding , then also / ' is an embedding. Now, let V' and V be the Levi-Civita connections of M' and M', respectively, and denote by B', C" and V'x the second fundamental tensors and the normal connection of the transversal mapping / ' of / . Then, from (1.3), (2.1) and (2.2), we get (3.1)

(B'(Xr, Y'))* = B(X'*, y"),

(3.2)

(C'v.X'y = Cu-X" - g(B(X'\ §), £/")£,

(3.3)

(V^U'Y = V%.U'"

for X', Y' G %~(M') and U' e %{MY. Moreover, from (2.6), (1.3), (3.1) and (3.3) we obtain (3.4)

(Vjr*, {Y'\ Z'*) =((yx,B')(X', Z'))' - g(HX'*, Yft)B^, Z") - g(HX'\ Z'*)B(^, Y'*).

From this immediately follows PROPOSITION 3.1. Let f be an invariant immersion. Then we have (i) / is r\-parallel if and only if f is parallel; (ii) / is totally geodesic if and only if f is totally geodesic. In the rest of this section we focus on congruent isometric immersions. DEFINITION 3.1. Let f,i = 1, 2, be tangent isometric immersions of {Mt, g,) into (M, g, &0. Then f\ and f2 are said to be A{M)-congruent or briefly, congruent, if there exists an element (p of A(M) and an isometry