Variations of the total mixed scalar curvature of a distribution

We examine the total mixed scalar curvature of a smooth manifold endowed with a distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution and use this key and technical result to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g., on the distribution). For each of those cases, we obtain the Euler–Lagrange equation and its several solutions. Examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows, codimension-one foliations, and distributions of interesting geometric properties (e.g., totally umbilical and minimal).


Introduction
Distributions on manifolds appear in various situations-e.g., as fields of tangent planes of foliations or kernels of differential forms. When the metric of a pseudo-Riemannian manifold is non-degenerate on a distribution, it defines a pseudo-Riemannian almost-product of general variations preserving the volume of the manifold. In Sect. 3.3, we consider the one-dimensional distribution spanned by the Reeb field on a contact manifold, which allows us to give an interpretation of some geometric quantities that appear in the Euler-Lagrange equation. Using the results obtained earlier for geodesic Riemannian flows, we show that the metrics of K -contact structures are critical with respect to all variations that fix the volume and partially preserve the metric (either on the distribution or everywhere else except it), thus generalizing a theorem from [4]. As a different application of the variational formulas obtained earlier, we also examine a measure of non-integrability of the orthogonal complement of the Reeb field, showing that contact metric structures are critical for this functional. The results we obtain for contact manifolds are then generalized to the setting of 3-Sasakian manifolds. Finally, in Sect. 3.4, we consider variation of the total mixed scalar curvature of a manifold endowed with a non-integrable distribution. Because of the complexity of the arising Euler-Lagrange equation, we look only for those critical metrics for which the orthogonal complement of the distribution is integrable. We show that K -contact and 3-Sasakian metrics (when the orthogonal complement of the distribution is integrable and has dimension one or three) are critical with respect to all variations that fix the volume and partially preserve the metric also in this setting. In case of codimension-one distribution, one of the variations that we consider has a particularly interesting geometric interpretation; we give an additional example of metric critical with respect to it.

Main results
In this part, we give necessary definitions, develop variation formulas for geometric quantities (that is the most technical and key result), and formulate the Euler-Lagrange equation for the total mixed scalar curvature of a manifold endowed with a distribution.

Preliminaries
This section recalls definitions of some functions and tensors, used also in [2,14] and introduces several new notions related to geometry of pseudo-Riemannian almost-product manifolds.
Let Sym 2 (M) be the space of all symmetric (0, 2)-tensors tangent to a smooth manifold M. A pseudo-Riemannian metric of index q on M is an element g ∈ Sym 2 (M) such that each g x (x ∈ M) is a non-degenerate bilinear form of index q on the tangent space T x M. For q = 0 (i.e., g x is positive definite), g is a Riemannian metric, and for q = 1 it is called a Lorentz metric. Let Riem(M) ⊂ Sym 2 (M) be the subspace of pseudo-Riemannian metrics of given signature.
Let [X,Y ] be the curvature tensor of the Levi-Civita connection ∇ of g. At a point x ∈ M, a two-dimensional linear subspace X ∧Y (called a plane section) of T x M is non-degenerate if W (X, Y ) := g(X, X ) g(Y, Y ) − g(X, Y ) g(X, Y ) = 0. For such section at x, the sectional curvature is the number K (X ∧ Y ) = g(R(X, Y )X, Y )/W (X, Y ).
A subbundle D ⊂ T M (called a distribution) is non-degenerate, if g x is non-degenerate on D x ⊂ T x M for every x ∈ M; in this case, the orthogonal complement of D, denoted by D, is also non-degenerate [10], and we have D x ∩ D x = 0, D x ⊕ D x = T x M for all x ∈ M. A connected manifold M n+ p with a pseudo-Riemannian metric g and a pair of complementary orthogonal non-degenerate distributions D and D of ranks dim D x = n and dim D x = p for every x ∈ M is called a pseudo-Riemannian almost-product structure on M, [8]. Such (M, D, D, g) is also sometimes called a pseudo-Riemannian almost-product manifold. Let Riem(M, D, D) ⊂ Riem(M) be the subspace of metrics making D and D orthogonal and non-degenerate.
Let X M be the module over C ∞ (M) of all vector fields on M, and let X D and X D be the modules of sections of D and D, respectively. The following convention is adopted for the range of indices: The "musical" isomorphisms and will be used for rank one and symmetric rank 2 tensors. For example, if ω ∈ T 1 0 M is a 1-form and X, Y ∈ X M , then ω(Y ) = g(ω , Y ) and X (Y ) = g(X, Y ). For (0, 2)-tensors A and B we have A, B = Tr g (A B ) = A , B .
The sectional curvature K (X ∧ Y ) is called mixed if X ∈ D and Y ∈ D. Let {E a , E i } be a local orthonormal frame adapted to ( D, D), i.e., E a ∈ D, E i ∈ D, and let i = g(E i , E i ), a = g(E a , E a ). We have | i | = | a | = 1 and W (E a , E i ) = a i = 0. The function on M, is called the mixed scalar curvature, see [18], and does not depend on the choice of the adapted orthonormal frame. If a distribution is spanned by a unit vector field N , i.e., g(N , N ) = N ∈ {−1, 1}, then S mix = N Ric N ,N , where Ric N ,N is the Ricci curvature in the N -direction.
To compute S mix on (M, g) we only need to fix one of the distributions, say D, then we obtain the second distribution as its g-orthogonal complement and the function (1) is well defined. Given a pair (M, D) of a manifold and a distribution, we shall study pseudo-Riemannian structures non-degenerate on D and critical for the functional where in (2) is a relatively compact domain of M (and = M when M is closed), containing supports of variations of the metric. The Euler-Lagrange equation for (2), that we shall obtain later, is expressed in terms of extrinsic geometry of the distribution D and its orthogonal complement D. In order to understand it, we shall define several notions on a pseudo-Riemannian almost-product manifold (M, D, D, g).
For every X ∈ X M we have X = X + X ⊥ , where X ≡ X is the D-component of X (respectively, X ⊥ is the D-component of X ) with respect to g. We define g ⊥ and g by The symmetric (0, 2)-tensor r D , given by is referred to as the partial Ricci tensor adapted for D; see [2,14]. In particular, by (1), Note that the partial Ricci curvature r D (X, X ) in the direction of a unit vector X ∈ D is the sum of sectional curvatures over all mixed planes containing X .
Let T, h : D × D → D andT ,h : D × D → D be the integrability tensors and the second fundamental forms of D and D, respectively.
Using an orthonormal adapted frame, one may find the formulae The mean curvature vector fields of D and D are, respectively, A distribution D is called totally umbilical, minimal, or totally geodesic, if h = 1 n H g , H = 0, or h = 0, respectively. There exist minimal, nowhere totally geodesic distributions of any codimension > 1 on Lie groups with left-invariant metrics, see [15]. In the case of foliations, the metric can be chosen to be bundle-like and mixed scalar curvature is leafwise constant.
The Weingarten operator A Z of D with respect to Z ∈ D, and the operator T Z are defined by For the local orthonormal frame {E i , E a } (adapted to the distributions), we use the following convention for various (1, 1)-tensors: The Divergence Theorem states that M (div ξ) d vol g = 0, when M is closed; this is also true if M is open and ξ ∈ X M is supported in a relatively compact domain ⊂ M.
Indeed, using H = a≤n a h(E a , E a ) and g(X, E a ) = 0, one derives (4): For a (1, 2)-tensor P define a (0, 2)-tensor div ⊥ P by Then the divergence of P is div P = div P +div ⊥ P. For a D-valued (1, 2)-tensor P, similarly to (4), we have where P, H (X, For any function f on M, we introduce the following notation of the projections of its gradient onto distributions D and D: The D-Laplacian of a function f is given by the formula As in [2,14], we define self-adjoint (1, 1)-tensors: A := i i A 2 i , called the Casorati operator of D, and T := i i (T i ) 2 . Similarly, we define A = a aÃ 2 a and T = a a (T a ) 2 . We also define the symmetric (0, 2)-tensors and by formulas The partial Ricci tensor can be presented in terms of the extrinsic geometry. Using its definition and the decomposition of tangent bundle into two orthogonal distributions, similarly as in [2], one can obtain the following lemma, that we prove below for readers' convenience.
Lemma 1 Let g ∈ Riem(M, D, D). Then the following identity holds: Proof For X, Y ∈ X D and U, V ∈ X D we have, see [11,Lemma 2.25], where the conullity tensorsC : D × D → D and C : D × D → D are defined bỹ . We can assume that ∇ X Y ∈ D x and ∇ X E a ∈ D x at a given point Let divC = n a=1 a (∇ aC ) a . Then, tracing (7) over D x yields Using Tr g (A Y T X ) = 0 = Tr g (T Y A X ) (since h is symmetric and T is antisymmetric), we extract (6) as the symmetric part of (8).
We define a self-adjoint (1, 1)-tensor (with zero trace) It is easy to see that all the above tensors defined with the use of an adapted orthonormal frame in fact do not depend on the choice of such frame.
Remark 1 (see [14]) Let us clarify the geometrical sense of h andK. If g is definite on D, then h = 0 if and only if one of the following point-wise conditions holds: If D is integrable, thenT a = 0 (a = 1, . . . , n), henceK := a a [T a ,Ã a ] = 0. If D is totally umbilical, then every operatorÃ a is a multiple of identity andK vanishes as well.

Variation formulas
Let (M, D, g) be a manifold with distribution and a pseudo-Riemannian metric g. We consider smooth 1-parameter variations {g t ∈ Riem(M) : |t| < ε} of the metric g 0 = g. We assume that the induced infinitesimal variations, represented by a symmetric (0, 2)-tensor B t ≡ ∂g t /∂t, are supported in a relatively compact domain in M, i.e., g t = g outside for all |t| < ε. We adopt the notations but we shall also write B instead of B t to make formulas easier to read, wherever it does not lead to confusion. Since B is symmetric, for any (0, 2)-tensor C, we have C, B = Sym(C), B . We denote by D(t) the g t -orthogonal complement of D.
Definition 1 (i) Let D be a distribution on (M, g). A family of metrics {g t ∈ Riem(M) : |t| < ε} such that g 0 = g and for all |t| < ε: will be called g ⊥ -variation. For g ⊥ -variations the metric on D is preserved. (ii) Let D be a distribution on (M, g) and let D be its g-orthogonal complement. A family of metrics {g t ∈ Riem(M) : |t| < ε} such that g 0 = g, for all |t| < ε the distributions D and D remain orthogonal and will be called g -variation. For g -variations only the metric on D changes.
We will now relate the variations defined above to arbitrary variations of g. Let D = D(0) be the g-orthogonal complement of D. While the distributions D and D may not be g t -orthogonal for t > 0, we can assume that they span the tangent bundle. For any X ∈ T M, let X D denote the g-orthogonal projection of X onto D and let X D denote the g-orthogonal projection of X onto D.
, and we can present g t in the following form: For g ⊥ -variations, g t = g t| D×D + g t | V + g 0 | D× D and for g -variations g t = g 0| D×D + g t | D× D (as g t | V = g 0 | V = 0), we have, respectively, By the above, the derivative B t of any variation g t can be decomposed into sum of derivatives of some g ⊥ -and g -variations.
For all X, Y, Z ∈ X M , the Levi-Civita connection ∇ t of g t (|t| < ε) evolves as, see for example [17], where the first covariant derivative of a (0, 2)-tensor B is expressed as Let D(t) be the g t -orthogonal complement of D. Let and ⊥ denote the g t -orthogonal projections onto D and D(t), respectively; note that these projections are t-dependent.
Then, for all t, , and for any X , we have g t (E i , X ) = 0. We can finish the proof by computing The evolution of D(t) gives rise to the evolution of both Dand D(t)-components of any vector X on M.

Lemma 3 Let g t be a g ⊥ -variation of g. Then for any t-dependent vector X on M, we have
Proof Using the frame from Lemma 2, we can write We have The proof follows from differentiating (12) and using (11) and (13).
Remark 2 Let B be a symmetric (0,2)-tensor. The following computations will be used to obtain variation formulas: Later we will also use the fact that for X ∈ D, N ∈ D we have Similar formulas can be obtained for α,α , θ,α , etc.
The key and most technical result of this section is the following.

Proposition 1 Let g t be a g ⊥ -variation of g. Then
Proof In the proof we denote by (i) j the jth term of the right-hand side of formula (i). We shall use an adapted frame that satisfies ∇ t X E j ∈ D and ∇ t X E b ∈ D for all X ∈ T x M, at a point x ∈ M for which all the formulas are considered, and for the value of parameter t at which the variation is computed. (All the results hold true without this assumption, but it simplifies the computations.) Proof of (14a). We use Lemma 3 to compute ∂ t h ,h , as a sum of 10 terms in g(· , E a ), The last two terms (15) 9 and (15) 10 are equal, and their sum can be computed in the following way: Using Lemma 2, we rewrite (15) as From the definition 2 Sym and we obtain (14a), using the following computations for all seven lines of (16): As an example, we give a detailed computation of the fourth line above: Note that whileα =α |V , its divergence divα may not vanish on D × D or D × D. Proof of (14b). We compute for any X ∈ T x M, using Lemmas 2 and 3, Using known formula (10) for t-derivative of the Levi-Civita connection, see [17], we present (17) 5 (i.e., the fifth term in g(·, X ) of (17)) as the following sum (we omit summation by i below): The last two terms above, (18) 5 and (18) 6 , vanish by the assumption We present the term (17) 3 as the sum of two terms and then rewrite the term (19) 2 as Note that (18) For the term (17) 1 , we get For the term (18) 3 , we obtain For the term (18) 4 , we get Finally, we collect results: Finally note that Tr D B = g, B , and that completes the proof of (14b). The computations for h and H are easier, since B(X, Y ) = 0 for X, Y ∈ D. Proof of (14c). We observe that where, using Lemma 3 and formula (10) for ∂ t ∇ t , we compute We used in the above due to which the last four terms in the formula above vanish. Note that Finally, we obtain (14c): Proof of (14d). We observe that For arbitrary X ∈ T x M, using Lemma 3 and formula (10) for ∂ t ∇ t , we obtain It follows that Finally, using B(H,H ) = Sym(H ⊗H ), B , we obtain (14d).
Proof of (14f). We calculate using Lemma 3, Then we obtain (14f): This completes the proof.
Similarly as Lemma 2, one can prove the following D)-adapted and g-orthonormal frame. For any variation g t , the frame evolving according to equations: where B = ∂ t g t , remains an orthonormal frame adapted to D and D(t).
For any g -variation, the frame evolving according to equations: where B = ∂ t g t , remains an orthonormal frame adapted to D and D.
Lemma 3 remains true without any changes for both g t -and g -variations, and Proposition 1 has the following analogue.

Proposition 2 For g -variations of g, we have
Proof The claim follows in fact from the computations that we already did in the proof of Proposition 1. Careful comparison of Lemmas 2 and 4 indicates that in order to obtain (24a)-(24f) it is enough to take formulas dual (with respect to interchanging D and D) to (14a)-(14f) and assume in them that B = B | D× D .
Remark 3 Note that g -variations coincide with one of two families of adapted variations considered in a previous paper of the authors [14] and in [2]. The adapted variations are a special case of variations considered in this paper-as they are additionally required to keep the distributions D and D(0) orthogonal for all g t . Here we make no such assumption, allowing the g t -orthogonal complement of the distribution D to vary, which enables us to consider arbitrary variations of the metric. Indeed, one can prove that variation formulas for general variations g t are sums of the corresponding formulas from Propositions 1 and 2. This follows from the fact that every infinitesimal variation of g can be decomposed into the sum of infinitesimal g ⊥ -and g -variations. Such decomposition would not be possible with the use of adapted variations only.
As the last of technical tools that we shall use, we note the following formula for variation of the volume form, true for any variation of a metric g t with B = ∂ t g t|t=0 [17]:

Euler-Lagrange equation
In this section, we present the Euler-Lagrange equation for the action (2). We consider different kinds of variations of metric. For arbitrary variations of the metric, the Euler-Lagrange equation is simply a condition for vanishing of the gradient of the functional: for any variation g t with B = ∂ t g t |t=0 . In analogue to the Einstein-Hilbert action, one can also consider variations preserving the volume of . For such variations, using (25), we have Hence, metric g is critical for the volume-preserving variations if and only if the condition δ J mix, D, , B d vol g = 0 holds for all B satisfying g, B = 0. It follows that the Euler-Lagrange equation is now where λ ∈ R is an arbitrary constant [3] (i.e., every metric satisfying it with some constant λ ∈ R is critical). Unfortunately, in our case the functional J mix, D, is not a Riemannian functional (i.e., it is not invariant under all diffeomorphisms of M), hence we cannot take as λ an arbitrary function [3]. Note that the Euler-Lagrange equation for arbitrary variations is a special case of (26), with λ = 0. We can also consider volume-preserving g -and g ⊥ -variations. For g -variations, B is restricted to D × D and for g ⊥ -variations B vanishes on D × D. Hence, the Euler-Lagrange equation is still (26), only either restricted to D× D (for g -variations) or considered everywhere except D × D (for g ⊥ -variations).

A metric g ∈ Riem(M, D, D) is critical for the action (2) with respect to volume-preserving g -variations if and only if
Proof Let g t be a g ⊥ -variation, and let Q(g) := S mix − div(H +H ). Then Differentiating the formula div X ·d vol g = L X (d vol g ), and using (25) we obtain ∂ t ( div X ) = div(∂ t X ) + (1/2) X (Tr B ) for any t-dependent vector field X . In particular, it follows that For g ⊥ -variations supported inside , it follows from (20) and (21) and Q(g) can be presented using (9) as Applying Corollary 1 and Proposition 1 to (28), using (5) and removing integrals of divergences of vector fields compactly supported in , we get where B = {∂ t g t } | t=0 . Since by (29) and (25), we have If g is critical for J mix, D, with respect to g ⊥ -variations, then the integral in (30) is zero for arbitrary symmetric (0, 2)-tensor B vanishing on D × D. This yields the Euler-Lagrange equation, that we can decompose into two independent parts: its D × D and V-components, obtaining the following: 2 θ,H + 2 α,θ + (div(α −θ)) |V + α,α+θ + θ −α, H For volume-preserving g ⊥ -variations, the Euler-Lagrange equation will be (31b), and instead of (31a), one needs to consider the following: Using tensor r D (Lemma 1) and replacing divh in (31a) according to (6), we rewrite (32) as (27a). Using the properties P,Q = Q,P and P,Q 1 +Q 2 = P,Q 1 + P,Q 2 , we rewrite (31b) as (27b). Finally, using the fact that all variation formulas for g -variations are dual to the D × D components of the variation formulas for g ⊥ -variations, we can take the dual equation to (32) to obtain the following Euler-Lagrange equation for volume-preserving g -variations: as the dual to S mix is S mix . Using the dual of Lemma 1 yields (27c). (27a)  In general, it is difficult to find critical points of (2) for arbitrary variations of metric. A trivial example of such metric is the one of the metric product of manifolds, i.e., with both D and D integrable and totally geodesic. A more interesting case (to be considered in further work) are critical left-invariant metrics on Lie groups endowed with left-invariant distributions. There exist, however, many interesting examples of metrics critical with respect to volume-preserving variations, or volume-preserving g ⊥ -and g -variations considered separately-we shall present some of them in further sections. Note that the volume-preserving g ⊥ -and g -variations generalize other variations considered in literature, e.g., the variation among associated metrics on a contact manifold [4], discussed in Sect. 3.3.

Particular cases
In this part of the paper, we examine the Euler-Lagrange equations (27a)-(27c), assuming particular (co)dimension of the distribution D or the existence of an additional structure on the manifold M. In these special geometric settings, we obtain examples of metrics critical for the action (2), with respect to variations previously discussed.

Flows
Let D be spanned by a nonsingular vector field N , and then it is tangent to the one-dimensional foliation by the flowlines of N . In this case, S mix = N Ric N ,N , R N = R(N , · )N is the Jacobi operator and the partial Ricci tensor takes a particularly simple form: We haveh =h sc N , whereh sc = N h , N is the scalar second fundamental form of D. Let A N be the Weingarten operator associated withh sc and letτ i = TrÃ i N (i ≥ 0). We have S ex = g(H, H ) − h, h = g(H, H ) − g(H, H ) The curvature of the flow lines is H = N ∇ N N . From Theorem 1, we obtain the following. Proof An easy computation shows that (H, H ), Let X be orthogonal to N with ∇ Z X ∈ D for all Z ∈ T M. We have θ = 0 and since

Corollary 2 (Euler-Lagrange equation) Let a distribution D be spanned by a unit vector field N on a manifold M with respect to g ∈ Riem(M, D, D). Then g is critical for the action (2) with respect to volume-preserving g ⊥ -variations if and only if
Hence, (36) is written as (34b).
By (5), we have divh = N (h sc ) −τ 1hsc and div h = (div H )g. Then, see (6) and (9), Remark that (37) 2 is simply the trace of (37) 1 . A flow of a unit vector N is called geodesic if the orbits are geodesics (h = 0) and Riemannian if the metric is bundle-like (h = 0). A nonsingular Killing vector field clearly defines a Riemannian flow; moreover, a Killing vector field of constant length generates a geodesic Riemannian flow. Restricting Corollary 2 to the case of a geodesic Riemannian flow, we obtain the following. (M p+1 , g). Then g is critical for the action (2) with respect to volume-preserving g ⊥ -variations if and only if all the following conditions hold: For a geodesic Riemannian N -flow, (34b) reduces to condition div ⊥T N (X ) = 0 for all X ∈ D, that we shall now examine. A Riemannian geodesic flow locally gives rise to a Riemannian submersion with totally geodesic fibers. Such mappings can be described by the following tensor, introduced by Gray [8] and adjusted here to our notation:

Corollary 3 Let D be spanned by a unit vector field N that generates a geodesic Riemannian flow on a pseudo-Riemannian manifold
Let X ∈ D and ∇ Z X ∈ D for all Z ∈ T M. Using an adapted frame with E i ∈ D at a point, the fact that ∇ N N = 0, and the antisymmetry of ∇ Z O for all Z ∈ T M, we obtain: From the formula (5.37e) from [16], adjusted to our definitions of R and Ric, it follows that Thus, we obtain (38b). Finally, for volume-preserving g -variations, we have the Euler-Lagrange Eq. (34c), that for geodesic Riemannian flows takes the form N Ric N ,N = − 2 3 λ.
From Corollary 3, we immediately obtain the following.

Corollary 4
Let (M p+1 , g), with p > 1, be an Einstein manifold with a geodesic Riemannian flow. Let D be the 1-dimensional distribution tangent to the flowlines. Then g is critical for the action (2) with respect to volume-preserving g ⊥ and g -variations.
The following proposition shows that the only manifolds with geodesic Riemannian flows critical for the action (2) with respect to all volume-preserving variations locally are in fact metric products.

Proposition 3 Let D be spanned by a unit vector field N that generates a geodesic Riemannian flow on a pseudo-Riemannian manifold (M p+1 , g). If g is a critical metric for the action (2) with respect to all volume-preserving variations then D is integrable.
Proof Using Remark 4(ii), we can write the Euler-Lagrange equation for arbitrary volumepreserving variations as follows: where λ ∈ R is an arbitrary constant. Using R N = − T and taking its trace we obtain (4 − p) T ,T = 2λp. It follows from n, p ≥ 0 that the two equations for λ have a solution only for T ,T = 0.

Codimension-one foliations
In this section, we consider the action (2), where D is tangent to a codimension-one foliation (of dimension n > 1). We find metrics critical with respect to volume-preserving g ⊥ -and g -variations, as well as arbitrary volume-preserving variations. Let F be a codimension-one foliation tangent to the distribution D. Let h sc be the scalar second fundamental form, and A N the Weingarten operator of F ; then we define the functions τ i = Tr A i N (i ≥ 0)-the power sums of the principal curvatures k i of the leaves. The τ 's can be expressed using the elementary symmetric functions σ 1 , . . . , σ n , called mean curvatures in the literature. For example, σ 0 = 1 = τ 0 , σ 1 = τ 1 , and 2σ 2 = τ 2 1 − τ 2 .
We have T = 0 =T and We define the vector field ( div A N ) ∈ X D by the following equation: Then we can formulate the following.

Proposition 4 Let D be the distribution tangent to a codimension-one foliation of a manifold M n+1 . Then a metric g on M is critical for the action (2) with respect to volume-preserving g ⊥ -variations if and only if
and g on M is critical for the action (2) with respect to volume-preserving g -variations if and only if:

Moreover, a metric g on M is critical for the action (2) with respect to all volume-preserving variations if and only if (40a)-(40c) hold with the same constant λ.
Proof Equations (40a) and (40c) follow from (27a) and (27c). This can be shown by a direct computation, but since the orthogonal complement of D is spanned by a single vector field N , we can use the equations obtained for flows, as adapted (i.e., with ∂ t g restricted to D×D) g ⊥variations in this section correspond to g -variations in Sect. 3.1. (See Remarks 3 and 4(i).) Hence, (40a) is dual to (34c) and (40c) is dual to (34a), with the additional assumption that D is integrable. Indeed, using (37) in (34c) and (34a), then taking their duals and setting T = 0, we obtain (40a) and (40c). For X ∈ D and N ∈ D, we have Using the above equations and the fact that for all X ∈ D we have we reduce (27b) to (40b).
From Proposition 4, we obtain the following.

Corollary 5 Let D be tangent to a codimension-one foliation of a pseudo-Riemannian manifold (M n+1 , g), and let the unit normal field N of F be complete in M. Then metric g is critical for the action (2) with respect to all volume-preserving variations if and only if
and τ 1 is bounded on M only for λ ≥ 0; moreover, τ 1 = 0 when λ = 0.
Using the above, one can obtain where f i (i = 1, . . . n) are positive functions. Then (41) is written as the system where τ 1 = y 1 + · · · + y n , τ 2 = y 2 1 + · · · + y 2 n , λ is a constant and g 00 is a smooth function of constant sign. Also, (42) takes the following form: Proof This follows from a straightforward computation, using (45) and Lemma 5.
The following lemma shows how the Euler-Lagrange Eqs. (46b) and (46a) are related, when the same constant λ is considered in both of them.
We use the last lemma to give a construction of metric of the form (45) with g 00 ≡ 1, that is critical for the action (2) with respect to arbitrary volume-preserving variations.
Then the metric g given by (45), with f i and y i as above, is critical for the action (2) with respect to arbitrary volume-preserving variations.
Example 1 Let n = 2, N = 1 and g 00 ≡ 1. Then we have the following solution of (46b) and (46a): where c is an arbitrary constant.
It is more difficult to find critical metrics of particular geometric properties.

Proposition 6
Let g be a metric on M n+1 , with n > 1, critical for the action (2) with respect to all volume-preserving variations and let D be tangent to a codimension-one foliation. If D is minimal, then D is totally geodesic.

Proposition 7
Let g be a metric on M n+1 , with n > 1, critical for the action (2) with respect to all volume-preserving variations and let D be tangent to a codimension-one foliation, with unit normal field N . If D is totally umbilical, then it has constant mean curvature τ 1 .

Contact and 3-Sasakian structures
Contact manifolds come with a natural foliation given by the flowlines of the Reeb field. They also admit an (in general, non-unique) associated metric of well examined properties; we show that for such metric one of the Euler-Lagrange Eq. (27b), always holds. We use this fact later to show that 3-Sasakian structures are a natural source of the metrics critical for the action (2). Recall [4] that a manifold M 2n+1 with a 1-form η such that is called a contact manifold, and ξ is called the characteristic vector field (or the Reeb field).
The above (φ, ξ, η, g) is called a contact metric structure on M. For all contact manifolds we consider in this section, let D be spanned by ξ and let D denote its orthogonal complement.
Remark 5 While we shall consider only the Riemannian metric in this section, there is a natural way to make a Riemannian contact manifold (M, η, g) a pseudo-Riemannian contact manifold: by settingḡ = g − 2 η ⊗ η as the new metric [6]. Then −ḡ(X, ξ) = η(X ) for all X ∈ T M and the remaining equations of (48) hold forḡ without changes. This transformation does not invalidate our main results.
On any contact manifold, there exists a (non-unique) contact metric structure; see [4]. Among them there is a class particularly interesting from the geometric point of view.

Definition 2 [4]
A contact metric structure for which ξ is Killing is called K -contact.

Proposition 9
Any K -contact metric g is critical for the action (51), with respect to both volume-preserving g ⊥ -and g -variations.
Proof We have already seen in (49) that the integral curves of ξ are geodesics for the contact metric structure. On the other hand, a nonsingular Killing vector field defines a Riemannian flow (h = 0). Thus, in case of a K -contact structure, we can use Corollary 3.
In [4], the action (2), which reduces to has been studied on the set of metrics associated with a given contact form.

Definition 3 [4, p. 24]
A contact structure is regular if ξ is regular as a vector field, that is, every point of the manifold has a neighborhood such that any integral curve of ξ passing through the neighborhood passes through only once. We have g(ξ, ξ ) = 1 for any associated metric and the volume form of associated metric on a contact manifold can be expressed only in terms of η and dη. Therefore, variations of the metric restricted to the set of all associated metrics form a subclass of the volumepreserving g ⊥ -variations. Hence, on compact regular contact manifolds, Proposition 9 and Theorem 2 together give the following characteristic of some critical metrics-for a larger space of variations.

Proposition 12
The metric of a Sasakian 3-structure on M is critical for the action (2) (where D is spanned by the characteristic vector fields), with respect to both volume-preserving g ⊥and g -variations.
Proof Since every ξ a defines a Sasakian structure, we can use the following formulas for any unit vectors X, Y orthogonal to ξ a (so we can also have X = ξ b , etc.): The above formulas are consistent with their analogues for ξ b and ξ c , and yield the following: r D = 3g ⊥ and r D = pg . We also have and T ,T = 3 p. It follows that (27a) and (27c) are satisfied, but never with the same constant λ. The remaining Euler-Lagrange Eq. (27b) reduces to (divθ) |V = 0. Since g is a K -contact metric for ξ a , for all Y ∈ T M we have ∇ Y ξ a = −φ a (Y ) [4] andT a (Y ) = φ a (Y ), and it follows that (divθ)(Y, ξ a ) = (div φ a )(Y ). As for any contact metric structure, we have ∇ ξ a ξ a = 0 and φ a (ξ a ) = 0. Any vector X ∈ D is orthogonal to ξ a and in all tensor formulas we can assume that ∇ Z X is colinear with ξ a for all Z ∈ T M. Then and similarly for ξ b , ξ c . The formula we obtained above, when considered for a contact metric structure (φ a , ξ a , η a , g) on M, is precisely (div ⊥ φ a )(X ). In the proof of Proposition 8 we showed that (50) holds, and hence (27b) is satisfied.

Non-integrable distributions
In this section we examine the action (2) for a fixed, non-integrable distribution D on a manifold (M, g). As there is no explicit procedure of solving the Euler-Lagrange equation in this case, we must resort to considering particular examples of critical metrics. For this purpose, we set as D the distribution orthogonal to the Reeb fields on contact and 3-Sasakian manifolds. In this setting, dual to the one considered in Sect. 3.3, K -contact and 3-Sasakian metrics are critical for the action (2) for volume-preserving g ⊥ -and g -variations. We show how a K -contact metric can be slightly modified and still remain critical. Finally, we consider g ⊥ -variations of a codimension-one distribution, and give an example of contact metric structure (that is not K -contact) critical with respect to them. that share the same transversal geometry. For example, if D is totally geodesic or umbilical, all foliations corresponding to metrics g t are Riemannian or conformal. Thus, g ⊥ -variation can be a tool to find the locally "best" (e.g., minimizing a functional) metrics for foliations of some fixed transverse property. Using equations dual to the ones formulated in Sect. 3.1 and some easy computations, we can obtain the following.

Proposition 14
Let D be a codimension one distribution on a manifold M n+1 with unit normal field N . A metric g on M is critical for the action (2) with respect to volume-preserving g ⊥ -variations if and only if: A metric g on M is critical for the action (2) with respect to volume-preserving g -variations if and only if: A metric g on M is critical for the action (2) with respect to all volume-preserving variations if and only if (55a)-(55c) hold, with the same constant λ.

Example 3
As an example, we can consider the following contact metric structure on R 3 , which is not K -contact [4]. Let Using an adapted orthonormal frame: E 1 = 2( ∂ ∂ x −z ∂ ∂ y +y ∂ ∂z ), E 2 = 2 ∂ ∂ y , E 3 = N = 2 ∂ ∂z , one can show that in {E 1 , E 2 } basis of D we havẽ We have Ric N ,N = 0 [4], is not conformal, hence (55c) is not satisfied, although (55a) holds. Further computations show that ∇ E 1 E 2 = −2E 3 and ∇ E 1 E 3 = 2E 2 are the only non-vanishing derivatives of vector fields E a from the frame, with respect to that frame. We obtain ( div A N ) = 0 and hence also (55b) is satisfied. It follows that g is critical for the action (2) with respect to volume-preserving g ⊥ -variations.