The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds, II

We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange equations (which in the case of space-time are analogous to those in Einstein–Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler–Lagrange equations of the mixed Einstein–Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections.


Introduction
We study the mixed Einstein-Hilbert action as a functional of two variables: a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution extends the original formulation of general relativity and provides interesting examples of metrics as well as connections, e.g., [1,Chapter 17]. Here, Λ is a constant (the "cosmological constant"), L is Lagrangian describing the matter contents, and a > 0 is the coupling constant. To deal also with non-compact manifolds, it is assumed that the integral above is taken over M if it converges; otherwise, one integrates over arbitrarily large, relatively compact domain Ω ⊂ M , which also contains supports of variations of g and T. The Euler-Lagrange equation for (2) when g varies is (called the Einstein equation) with the non-symmetric Ricci curvature Ric and the asymmetric energy-momentum tensor Ξ (generalizing the stress tensor of Newtonian physics), given in coordinates by Ξ μν = −2 ∂L/∂g μν + g μν L. The Euler-Lagrange equation for (2) when T varies is an algebraic constraint with the torsion tensor S of∇ and the spin tensor s c μν = 2 ∂L/∂T c μν (used to describe the intrinsic angular momentum of particles in spacetime, e.g., [30]): S(X, Y ) + Tr(S(·, Y ) − S(X, ·)) = a s(X, Y ), X,Y ∈ X M . (3b) Since S(X, Y ) = T X Y − T Y X, (3b) can be rewritten using the contorsion tensor. The solution of (3a,b) is a pair (g, T), satisfying this system, where the pair of tensors (Ξ, s) (describing a specified type of matter) is given. In vacuum space-time, Einstein and Einstein-Cartan theories coincide. The classification of solutions of (3a,b) is a deep and largely unsolved problem [30], even for T = 0 [7].

Objectives
On a manifold equipped with an additional structure (e.g., almost product, complex or contact), one can consider an analogue of (2) adjusted to that structure. In pseudo-Riemannian geometry, it may mean restricting g to a certain class of metrics (e.g., conformal to a given one, in the Yamabe problem [7]) or even constructing a new, related action (e.g., the Futaki functional on a Kahler manifold [7], or several actions on contact manifolds [8]), to cite only few examples. The latter approach was taken in authors' previous papers, where the scalar curvature in the Einstein-Hilbert action on a pseudo-Riemannian manifold was replaced by the mixed scalar curvature of a given distribution or a foliation. In this paper, a similar change in (2) will be considered on a connected smooth (n + p)-dimensional manifold M endowed with an affine connection and a smooth n-dimensional distribution D (a subbundle of the tangent bundle T M). Distributions and foliations (that can be viewed as integrable distributions) on manifolds appear in various situations, e.g., [5,20]. When a pseudo-Riemannian metric g on M is non-degenerate along D, it defines the orthogonal p-dimensional distribution D such that both distributions span the tangent bundle: T M = D ⊕ D and define a Riemannian almost-product structure on (M, g), e.g., [15]. From a mathematical point of view, a space-time of general relativity is a (n + 1)-dimensional time-oriented (i.e., with a given timelike vector field) Lorentzian manifold, see [4]. A space-time admits a global time function (i.e., increasing function along each future directed nonspacelike curve) if and only if it is stable causal; in particular, a globally hyperbolic spacetime is naturally endowed with a codimension-one foliation (the level hypersurfaces of a given time-function), see [6,12].
The mixed Einstein-Hilbert action on (M, D), is an analog of (2), where S is replaced by the mixed scalar curvature S mix , see (9), for the affine connection∇ = ∇ + T. The physical meaning of (4) is discussed in [2] for the case of T = 0. Our action (4) can be useful for the multitime Geometric Dynamics, e.g., [17] and survey [31]. This was introduced like Multi-time World Force Law involving field potentials, gravitational potentials (components of the two Riemannian metrics), and the Yang-Mills potentials (components of the Riemannian connections and the nonlinear connection).
In view of the formula S = 2S mix + S + S ⊥ , where S and S ⊥ are the scalar curvatures along the distributions D and D, one can combine the actions (2) and (4)  The mixed scalar curvature (being an averaged mixed sectional curvature) is one of the simplest curvature invariants of a pseudo-Riemannian almost-product structure. If a distribution is spanned by a unit vector field N , i.e., N, N = N ∈ {−1, 1}, then S mix = N Ric N,N , where Ric N,N is the Ricci curvature in the N -direction. If dim M = 2 and dim D = 1, then obviously 2S mix = S. If T = 0 then S mix reduces to the mixed scalar curvature S mix of ∇, see (10), which can be defined as a sum of sectional curvatures of planes that non-trivially intersect with both of the distributions. Investigation of S mix led to multiple results regarding the existence of foliations and submersions with interesting geometry, e.g., integral formulas and splitting results, curvature prescribing and variational problems, see survey [23]. The trace of the partial Ricci curvature (rank 2) tensor r D is S mix , see Sect. 2. The understanding of the mixed curvature, especially, r D and S mix , is a fundamental problem of extrinsic geometry of foliations, see [20].
Varying (4) with fixed T = 0, as a functional of g only, we obtain the Euler-Lagrange equations in the form similar to (3a), see [2] for space-times, and for D of any dimension, see [25,26], i.e., Considering variations of the metric that preserve the volume of the manifold, we can also obtain the Euler-Lagrange equations for (6), that coincide with those for (4) with L = 0 and Λ = 0. The terms ofS mix without covariant derivatives of T make up the mixed scalar T-curvature, see Sect. 2, which we find interesting on its own. In particular, S T can be viewed as the Riemannian mixed scalar curvature of a distribution with all sectional curvatures of planes replaced by their T-curvatures (see [18]), and for statistical connections we haveS mix = S mix + S T . Thus, we also study (in Sect. 3.1) the following, closely related to (6), action on (M, D): For each of the examined actions (6) and (7), we obtain the Euler-Lagrange equations and formulate results about existence and examples of their solutions, that we describe in more detail further below. In particular, from [27] we know that if T is critical for the action (6), then D and D are totally umbilical with respect to ∇-and to express this together with other conditions, a pair of equations like (3a,b) is not sufficient. Due to this fact, only in the special case of semi-symmetric connections we present the Euler-Lagrange equation in the form, which directly generalizes (5): and a separate condition (61), similar to (3b), for the vector field parameterizing this type of connection. In general case, instead of a single equation like (3b), we obtain a system of equations (30a-h), which we then use to write the analogue of (3a) explicitly in terms of extrinsic geometry of distributions.
In Sect. 3.3 we show that for n, p > 1 the critical value of (6) attained by (g, T), where T corresponds to a metric connection, depends only on g and is non-negative on a Riemannian manifold. In other words, pseudo-Riemannian geometry determines the mixed scalar curvature of any critical metric connection. For general metric connections, we consider only adapted variations of the metric (see Definition 2) due to complexity of the variational formulas. Compared to (6) with fixed g, we get a new condition (47a), involving the symmetric part of T| D×D and of T| D× D in the dual equation. This condition is strong enough to prevent existence of critical points of (6) in some settings, e.g. for D spanned by the Reeb field on a closed contact manifold with associated metric. Under some assumptions, trace of (47a) depends only on the pseudo-Riemannian geometry of (M, g, D) and thus gives a necessary condition for the metric to admit a critical point of (6) in a large class of connections (e.g., adapted), or for integrable distributions D. On the other hand, in the case of adapted variations, antisymmetric parts of (T| D×D ) ⊥ and (T| D× D ) remain free parameters of any critical metric connection, as they do not appear in Euler-Lagrange equations (note that these components define part of the critical connection's torsion). Thus, for a given metric g that admits critical points of (6), one can expect to have multiple critical metric connections, and examples in Sect. 3.3 confirm that. Section 3.4 deals with a semi-symmetric connection (parameterized by a vector field), as a simple case of a metric connection. Although such connections are critical for the action (6) and arbitrary variations of connections only on metric-affine products, when we restrict variations of the mixed scalar curvature to semi-symmetric connections, we obtain meaningful Euler-Lagrange equations (in Theorem 6), which allow us to explicitly present the mixed Ricci tensor-analogous to the Ricci tensor in the Einstein equation.

Preliminaries
Here, we recall definitions of some functions and tensors, used also in [3,[25][26][27]32], and introduce several new notions related to geometry of (M, g,∇) endowed with a non-degenerate distribution.

The Mixed Scalar Curvature
Let Sym 2 (M ) be the space of symmetric (0, 2)-tensors tangent to a smooth connected manifold M . A pseudo-Riemannian metric g = ·, · 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 a given signature.
The following convention is adopted for the range of indices: All the quantities defined below with the use of an adapted orthonormal frame do not depend on the choice of this frame. We have Definition 1. The function on (M, g,∇) endowed with a non-degenerate distribution D, is called the mixed scalar curvature with respect to connection∇. In particular case of the Levi-Civita connection ∇, the function on (M, g), is called the mixed scalar curvature (with respect to ∇). The symmetric (0, 2)tensor is called the partial Ricci tensor related to D.
Remark that on (M, D), the S mix and g-orthogonal complement to D are determined by the choice of metric g. In particular, if dim D = 1 then We use the following convention for components of various (1, 1)-tensors in an adapted orthonormal frame {E a , E i }: T a = T Ea , T i = T Ei , etc. Following the notion of T-sectional curvature of a symmetric (1, 2)-tensor T on a vector space endowed with a scalar product and a cubic form, see [18], we define the Vol. 76 (2021) The Mixed Scalar Curvature Page 9 of 56 162 mixed scalar T-curvature by (12), as a sum of T-sectional curvatures of planes that non-trivially intersect with both of the distributions, The definitions (12), (9)-(10) do not depend on the choice of an adapted local orthonormal frame. Thus, we can considerS mix and S T on (M, D) as functions of g and T. If T is either symmetric or anti-symmetric then (12) reads As was mentioned in the Introduction, the mixed scalar T-curvature (for the contorsion tensor T) is a part ofS mix , in fact we have [27,Eq. (6)]: whereQ consists of all terms with covariant derivatives of T, The formulas for the mixed scalar curvature in the next two propositions are essential in our calculations. The propositions use tensors defined in [25], which are briefly recalled below.

Proposition 2.
(see [21]) We have using (1), The tensors used in the above results (and other ones) are defined below for one of the distributions (say, D; similar tensors for D are denoted using or notation).
The integrability tensor and the second fundamental form T, h : D × D → D of D are given by The mean curvature vector field of D is given by H = Tr g h = a a h(E a , E a ). We call D totally umbilical, minimal, or totally geodesic, if h = 1 n H g , H = 0, or h = 0, respectively.
The "musical" isomorphisms and will be used for rank one and symmetric rank 2 tensors. For example, if ω ∈ Λ 1 (M ) is a 1-form and X, Y ∈ X M then ω(Y ) = ω , Y and X (Y ) = X, Y . For arbitrary (0,2)-tensors A and B we also have A, B = Tr g (A B ) = A , B .
The Weingarten operator A Z of D with Z ∈ X ⊥ , and the operator T Z are defined by The norms of tensors are obtained using The divergence of a vector field X ∈ X M is given by where d vol g is the volume form of g. One may show that The D-divergence of a vector field X is given by div For a (1, 2)-tensor P define a (0, 2)-tensor div ⊥ P by For a D-valued (1, 2)-tensor P , similarly to (21), we have where P, H is a (0, 2)-tensor, P, H (X, Y ) = P (X, Y ), H . For example, div ⊥ h = div h + h, H . For a function f on M , we use the notation ∇ ⊥ f = (∇f ) ⊥ of the projection of ∇f onto D.
The D-deformation tensor Def D Z of Z ∈ X M is the symmetric part of ∇Z restricted to D, The self-adjoint (1, 1)-tensors: A (the Casorati type operator ) and T and the symmetric (0, 2)-tensor Ψ , see [3,25], are defined by For readers' convenience, we gather below also definitions of all other basic tensors that will be used in further parts of the paper. We define a self-adjoint (1, 1)-tensor K by the formula and the (1, 2)-tensors α, θ andδ Z (defined for a given vector field Z ∈ X M ) on (M, D, g): For any (1, 2)-tensors P, Q and a (0, 2)-tensor S on T M, define the following (0, 2)-tensor Υ P,Q : where on the left-hand side we have the inner product of (0, 2)-tensors induced by g, {e λ } is a local orthonormal basis of T M and λ = e λ , e λ ∈ {−1, 1}. Note that Finally, for the contorsion tensor and X ∈ T M we define T X :D →D by Remark 1. From now on, we shall omit factors μ in all expressions with sums over an adapted frame (or its part), effectively identifying symbols μ with μ μ etc. As we assume in this paper that g is non-degenerate on the distribution D, the presence of factors μ in the sums is the only difference in formulas with adapted frames for a Riemannian and a pseudo-Riemannian metric g. With the definitions given in this section, all tensor equations that follow look exactly the same in both these cases. In more complicated formulas we shall also omit summation indices, assuming that every sum is taken over all indices that appear repeatedly after the summation sign and contains appropriate factors μ .

The Mixed Ricci Curvature
Let (M, g) be a pseudo-Riemannian manifold endowed with a non-degenerate distribution D. We consider smooth 1-parameter variations {g t ∈ Riem(M ) : |t| < } of the metric g 0 = g. Let the infinitesimal variations, represented by a symmetric (0, 2)-tensor be supported in a relatively compact domain Ω in M , i.e., g t = g outside Ω for all |t| < . We call a variation g t volume-preserving if Vol(Ω, g t ) = Vol(Ω, g) for all t. We adopt the notations ∂ t ≡ ∂/∂t, B ≡ ∂ t g t | t=0 =ġ, 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, then C, B = Sym(C), B for any (0, 2)-tensor C. We denote by ⊗ the product of tensors and use the symmetrization operator to define the symmetric product of tensors:

Definition 2.
A family of metrics {g t ∈ Riem(M ) : |t| < } such that g 0 = g will be called (i) g -variation if g t (X, Y ) = g 0 (X, Y ) for all X, Y ∈ X and |t| < .
(ii) adapted variation, if the g t -orthogonal complement D t remain g 0orthogonal to D for all t.
(iii) g -variation, if it is adapted and g t (X, Y ) = g 0 (X, Y ) for all X, Y ∈ X ⊥ and |t| < .
(iv) g ⊥ -variation, if it is adapted g -variation.
In other words, for g -variations the metric on D is preserved. For adapted variation we have g t ∈ Riem(M, D, D) for all t. For g -variations only the metric on D changes, and for g ⊥ -variations only the metric on D changes, and D remains to be g t -orthogonal to D.
The symmetric tensor B t =ġ t (of any variation) can be decomposed into the sum of derivatives of g -and g -variations, see [26]. Namely, Thus, for g -variations B(X, Y ) = 0 for all X, Y ∈ X . Denote by and ⊥ the g t -orthogonal projections of vectors onto D and D(t) (the g t -orthogonal complement of D), respectively. Proposition 3. (see [26]) Let g t be a g -variation of g ∈ Riem(M, D, D). Let {E a , E i } be a local ( D, D)-adapted and orthonormal for t = 0 frame, that evolves according to Then, for all t, {E a (t), E i (t)} is a g t -orthonormal frame adapted to ( D, D(t)).
For any g -variation of metric the evolution of D(t) gives rise to the evolution of both Dand D(t)-components of any X ∈ X M : when M is closed (compact and without boundary); this is also true if M is open and X is supported in a relatively compact domain Ω ⊂ M . For any variation g t of metric g on M with B = ∂ t g we have e.g., [29]. By Lemma 1 and (23)- (24), for any variation g t of metric with supp (∂ t g) ⊂ Ω, and t-dependent X ∈ X M with supp (∂ t X) ⊂ Ω. Definition 3. (see [22]) The symmetric (0, 2)-tensor Ric D in (5), defined by its restrictions on three complementary subbundles of T M × T M, is referred to as the mixed Ricci curvature: Here ( The following theorem, which allows us to restore the partial Ricci curvature (26), is based on calculating the variations with respect to g of components in (15) and using (25) for divergence terms. According to this theorem and Definition 3 we conclude that a metric g ∈ Riem(M, D) is critical for the action (6) with fixed T = 0 (i.e., considered as a functional of g only), with respect to volume-preserving variations of metric if and only if (5) holds. Theorem 1. (see [26]) A metric g ∈ Riem(M, D) is critical for the action (6) with fixed T = 0, with respect to volume-preserving g -variations if and only if Example 1. For a space-time (M p+1 , g) endowed with D spanned by a timelike unit vector field N , the tensor Ric D , see (26) with n = 1, and its trace have the following particular form: Hereτ i = Tr(( A N ) i ), A N is the shape operator, T is the integrability tensor and h sc is the scalar second fundamental form of D. Note that the right-hand side of (28) 2 vanishes when D is integrable.

Variations with Respect to T
The next theorem is based on calculating the variations with respect to T of components S T andQ/2 in (13) and using (25) .
for a one-parameter family T t (|t| < ε) of (1, 2)tensors. Using Proposition 2 and removing integrals of divergences of compactly supported (in a domain Ω) vector fields, we get Since no further assumptions are made about S or T, all the components S μ e λ , e ρ are independent and the above formula gives rise to (30a-h), where X, Y, Z ∈ D and U, V, W ∈ D are any vectors from an adapted frame. Observe that in every equation from (30a-h) each term contains the same set of those vectors and is trilinear in them, so all these equations hold in fact for all vectors X, Y, Z ∈ D and U, V, W ∈ D. Further below, we obtain many other formulas from computations in adapted frames, in the same way.
Taking difference of symmetric (in X, Y ) parts of (30c,e) with s = 0 yields that D is totally umbilical-similar result for D follows from dual equations (e.g., [27]). For vacuum space-time (L = 0), the (30a-h) are simplified to the following equations (31a-j).
of Example 1, the system (30a-h) reduces to

Main Results
In Sect. 3.1 we consider the total mixed scalar curvature of contorsion tensor for general and particular connections, e.g., metric and statistical, and metricaffine doubly twisted products. In Sect. 3.2 we consider the total mixed scalar curvature of statistical manifolds endowed with a distribution. In Sect. 3.3 we consider the total mixed scalar curvature of Riemann-Cartan manifolds endowed with a distribution. In Sect. 3.4, we derive the Euler-Lagrange equations for semi-symmetric connections and present the mixed Ricci tensor explicitly in (64). Our aims are to find out which metrics admit critical points of examined functionals and which components of T in these particular cases determine whether or not its mixed scalar curvature is critical in its class of connections. This might help to achieve better understanding of both mixed scalar curvature invariant and the role played by some components of contorsion tensor.

Variational Problem with Contorsion Tensor
By Proposition 2 and (12), we have the following decomposition [21] (note that these are terms of −Q in the first line of (18)): We consider arbitrary variations T(t), T(0) = T, |t| < , and variations corresponding to metric and statistical connections, while Ω ⊂ M contains supports of infinitesimal variations ∂ t T(t). In such cases, the Divergence Theorem states that if X ∈ X M is supported in Ω then (23) holds. and Moreover, if n > 1 and p > 1 then (33c,d) read as Proof. From Proposition 2 and Lemma 3, for a g -variation g t of metric g we obtain Thus, ∂ t S T (g t ) = 0 if and only if the right hand side of (35) vanishes for all but since B is arbitrary and symmetric and T * i E j − T * j E i is skew-symmetric, this can be written as (32a). For the mixed part of B (i.e., B restricted to the subspace V ) we get the following Euler-Lagrange equation: From this we obtain (32b). Taking dual equation to (32a) with respect to interchanging distributions D and D, we obtain (32c), which is the Euler-Lagrange equation for g -variations. The proof of (33a-g), see [27], is based on calculation of variations with respect to T of S T and using (25). [27]) The doubly twisted product B × (v,u) F of metric-affine manifolds (B, g B , T B ) and (F, g F , T F ) (or the metric-affine doubly twisted product ) is a manifold M = B × F with the metric g = g + g ⊥ and the affine connection, whose contorsion tensor is From Theorem 3 we obtain the following

respect to all variations of T and g if and only if
Tr Vol. 76 (2021) The Mixed Scalar Curvature Page 19 of 56 162 Proof. It was proven in [27,Corollary 13] that a metric-affine doubly twisted product B × (v,u) F is critical for (7) with fixed g and for variations of T if and only if (36) holds. It can be easily seen that for such doubly twisted product satisfying Tr T B = 0 = Tr T F all terms in (32a-c) vanish.
, where T is the contorsion tensor of a statistical connection on (M, g), is critical for the action (7) with respect to all variations of metric, and variations of T corresponding to statistical connections if and only if T satisfies the following algebraic system: Proof. By [27,Corollary 7], T is critical for the action T → M S T d vol g , see (7), with respect to variations of T corresponding to statistical connections if and only if the following equations hold: for all X, Y ∈ D and U, V ∈ D. If (37a,b) hold, then also (38a-c) hold, moreover if (37b) is satisfied and T corresponds to a statistical connection, then all terms in equations (32a-c) vanish.
If n > 1 we can similarly use (32c) for the same effect, and if n = p = 1 then (39) becomes which together with its dual imply (Tr T) = 0 = (Tr ⊥ T) ⊥ , and again we obtain (37b) from (38b,c).

Corollary 4. A pair (g, T)
, where T is the contorsion tensor of a metric connection on (M, g), is critical for (7) with respect to all variations of metric, and variations of T corresponding to metric connections if and only if T satisfies the following linear algebraic system (for all X, Y ∈ D and U, V ∈ D): and for all X ∈ D and U ∈ D we have Proof. By [27,Corollary 8], T is critical for the action T → M S T d vol g , see (7), with respect to variations of T corresponding to metric connections if and only if (40a-d) hold. In By (40d), this is identity if p > 1. On the other hand, for p = 1 it reduces to and by (40b), T 1 E 1 , E 1 = 0. Therefore, (32a) is satisfied if (40a-c) and the second equation in (40d) are satisfied. Using dual parts of (40a-d) we obtain analogous result for (32c). From (40a-d) we have for all b, c, i, k, Thus, in (32b) we have only the following terms: for all b, i. This completes the proof.
The results obtained when considering the action (7) on metric-affine doubly twisted products, allow us to determine which of these structures are critical for the action (6).

Proposition 4.
A metric-affine doubly twisted product B × (v,u) F is critical for (6) with respect to all variations of g and T if and only if (36) holds and Proof. It was proven in [27] that a metric-affine doubly twisted product B× (v,u) F is critical for action (6)

Statistical Connections
We define a new tensor Θ = T−T * +T ∧ −T * ∧ , composed of some terms appearing in (18).  Proof. For any T that corresponds to a statistical connection, we have T ∧ = T and T * = T. Condition 1 follows from (31a,f) and T = T ∧ . Then (31a,f), condition 1 and a, yield condition 3. We get condition 5 from T = T * and (31b). Conditions 4 and 6 are dual to conditions 3 and 5, and are obtained analogously. Condition 2 follows from T = T * , condition 3 (and its dual condition 5) and (31c) (and its dual (31g)). Condition 7 follows from Corollary 1.
Let g t be a g -variation of g. Although for statistical manifolds, (17) reads asS we cannot vary this formula with respect to metric with fixed T, because when g changes, T may no longer correspond to statistical connections (condition T = T * may not be preserved by the variation). Instead, we use Lemma 3 and derive from (67) for T corresponding to a statistical connection (for which For totally umbilical distribution, the last equation further simplifies to From conditions 2-4 we obtain in the above ∂ t Θ, A = 0. For integrable distributions, since Θ = 0, we have and from (71), with Θ = 0 and totally umbilical distributions, we have From conditions 3-4 and 2 we get in the above From conditions 3-4, using (72) and (73), we get From conditions 3-4 and 2 we get ∂ t Tr T * , Tr ⊥ T = 0. From T * = T, using (74), we obtain

From conditions 3-4 and 2 we get
From conditions 3-4 and 2 we get in the above By condition 6 we have H = 0 if p > 1 and if p = 1 we only have i = j = k = 1 and by condition 2, Hence, for T corresponding to a statistical connection satisfying the assumptions, any variation of S mix with respect to g is just a variation of S mix with respect to g. Thus, remaining (42a-c) are equations of Theorem 1 written for both distributions integrable and umbilical. Note that conditions for a statistical connection to be critical for (6) with fixed g are actually those from Corollary 3 (instead of conditions in [27, Theorem 3], which do not consider all symmetries of ∂ t T for variation among statistical connections). Indeed, for a family of statistical connections on (M, g) and . Gathering together terms appearing by these components in [27,Eq. (14)], we obtain Considering this, instead of [27, Eq. (28c)] we obtain the following Euler-Lagrange equation: which can be transformed into the third equation in [27,Cor. 7], and the second equation in [27,Cor. 7] is dual to it with respect to interchanging distributions D and D. Similarly, for terms appearing in [27,Eq. (14)] by g(S a E b , E c ), we obtain Considering this symmetry, we obtain the following Euler-Lagrange equation: and considering arbitrary a = b = c we get (Tr ⊥ T) = 0, which -together with its dual -is the first equation in [

Metric Connections
Here, we consider g and T as independent variables in the action (6), hence for every pair (g, T) critical for (6) the contorsion tensor T must be critical for (6) with fixed g, and thus satisfy Corollary 1. Using this fact, we characterize those critical values of (6), that are attained on the set of pairs (g, T), where T is the contorsion tensor of a metric (in particular, adapted) connection for g. Note that by [27,Corollary 2], restricting variations of T to tensors corresponding to a metric connection gives the same conditions as considering variations among arbitrary T.

Proposition 5.
Let the contorsion tensor T of a metric connection∇ be critical for the action (6) with fixed g. Then D and D are both totally umbilical and for Q given in (18) we have Proof. By Corollary 1, both distributions are totally umbilical. In this case, using (31a-j), we have Using the above in (18), and simplifying the expression, completes the proof.  Thus, the right hand side of the above equation is the only critical value of the action (6) (with fixed g on a closed manifold M ) restricted to metric connections for g. Notice that it does not depend on T, but only on the pseudo-Riemannian geometry of distributions on (M, g). Moreover, on a Riemannian manifold it is always non-negative.
Consider pairs (g, T), where T corresponds to a metric connection, critical for (6) with respect to g ⊥ -variations. We apply only adapted variations, as they will allow to obtain the Euler-Lagrange equations without explicit use of adapted frame or defining multiple new tensors. The case of general variations, mostly due to complicated form of tensor F defined by (66) that appears in variation formulas, is significantly more involved and beyond the scope of this paper. Set T satisfies the following linear algebraic system: Proof. By Corollary 1, T is critical for (6) (with fixed g) if and only if distributions D and D are totally umbilical and (47b-g) (together with (47h) if their respective assumptions on n and p hold) are satisfied. Let T be critical for the action (6) with fixed g. We shall prove that a pair (g, T) is critical for the action (6) with respect to g ⊥ -variations of metric if and only if (47a) holds. By Proposition 2, for any variation g t of metric such that supp(B) ⊂ Ω, where Ω is a compact set on M , and Q in (18), we have where X = (Tr (T − T * )) ⊥ + (Tr ⊥ (T − T * )) . Although X is not necessarily zero on ∂Ω, we have supp (∂ t X) ⊂ Ω, thus, d dt M (div X) d vol g = 0, see (25), and hence: where, up to divergence of a compactly supported vector field, ∂ t Q is given in Lemma 5. For g ⊥ -variations we get (see [26,Eq. (29)] for more general case of g -variations), For totally umbilical distributions we have Hence, where δQ is defined by the equality δQ, B = −∂ t Q, see Lemma 5. Thus, the Euler-Lagrange equation for g ⊥ -variations of metric and totally umbilical distributions is the following: By (47g), from the above we get (47a).

Remark 3.
Note that for volume-preserving variations, the right hand sides of (47a) and (48) should be λ g ⊥ , with λ ∈ R being an arbitrary constant [26]. This obviously applies also to the special cases of the Euler-Lagrange equation (47a) discussed below. If p > 1 and n > 1 then (47a) can be written as Taking trace of (49) and using (47d, g-i) and equalities Tr g Υ T,T = 2 T, T and Tr g T = − T ,T , we obtain the following result. a pair (g, T), where g is a pseudo-Riemannian metric on M and T corresponds to a metric connection, be critical for (6) with respect to g ⊥ -variations of metric and arbitrary variations of T. Then for n, p > 1 we have

Corollary 7. Let
and for n = 1 and p > 1 we get  Since g is definite on D, we obtain H = 0 and since D is totally umbilical by Theorem 5, the first claim follows; the second is analogous.
Recall that an adapted connection to (D, D), see e.g., [5], is defined bȳ and an example is the Schouten-Van Kampen connection with contorsion tensor

Proposition 6. Let D and D both be totally umbilical. Then contorsion tensor T corresponding to an adapted metric connection satisfies (47a-i) if and only if it satisfies the equations
for all X ∈ D and U ∈ D.
Proof. For adapted connection and totally umbilical distribution D we have φ = −2h = − 2 pH g ⊥ , see [27,Section 2.5], and Moreover, an adapted connection is critical for (6) with fixed g if and only if (51a-d) hold, see [27]. Note that for adapted connection from (52) we obtain aT a X, Y for umbilical distributions. Also (47h) hold, in all dimensions n, p. Thus, for a critical adapted connection, (47a) simplifies to (51e).
If p > 1 then φ ⊥ is not determined by (Tr ⊥ T) ⊥ and by (52) in Proposition 6 can be set arbitrary for an adapted metric connection. Using this fact and taking trace of (51e) yield the following.

Corollary 9. Let D and D both be totally umbilical. If a contorsion tensor
T, corresponding to an adapted metric connection, satisfies (47a-i) then the metric g satisfies Example 3. In [13] it was proved that on a Sasaki manifold (M, g, ξ, η) (that is, M with a normal contact metric structure [8]) there exists a unique metric connection with a skew-symmetric, parallel torsion tensor, and its contorsion tensor is given by T X Y, Z = 1 2 (η ∧ dη)(X, Y, Z), where X, Y, Z ∈ X M and η is the contact form on M . Let D be the one-dimensional distribution spanned by the Reeb field ξ. It follows that for this connection we have φ = 0 and for X, Y ∈ D , see (46), as dη(X, Y ) = 2 X,T ξ Y (we use here the same convention for differential of forms as in [13], which is different than the one in [8]). Since g is a Sasaki metric, both distributions are totally geodesic, and for volumepreserving variations the Euler-Lagrange equation (47a) gets λ g ⊥ on the righthand side (see Remark 3) and becomes As on a Sasakian manifolds we have T = − 1 p T ,T g ⊥ and T ,T = p (e.g., [26,Section 3.3]), we see that (54) holds in this case for λ = 5.
We can slightly modify this example to obtain a family of critical connections (although no longer with parallel torsion) on a contact manifold. Proposition 7. Let (M, η) be a contact manifold and let D be the one-dimensional distribution spanned by the Reeb field ξ. Let g be an associated metric [8] on (M, η).
1. There exist metric connections ∇ + T on M such that (g, T) is critical for (6) with respect to volume-preserving g ⊥ -variations of metric and arbitrary variations of T.

If M is closed, then there exist no metric connections ∇ + T on M such that (g, T) is critical for (6) with respect to adapted volume-preserving variations of metric and arbitrary variations of T.
Proof. 1. Let T ξ ξ = 0 and for all X, Y ∈ D let (T ξ X) = 0, (T X ξ) = 0 and where ω is any 3-form. Then connection ∇ + T will satisfy all Euler-Lagrange equations (47b-i) and (47a) with λg ⊥ = 5g ⊥ on the right-hand side (see Remark 3). 2. Recall [26,Remark 4(ii)] that the Euler-Lagrange equations for volumepreserving adapted variations of the metric are (47a) with the right-hand side λg ⊥ and its dual (with the same constant λ). Note that the tensor dual to φ is given byφ( For an an associated metric g, (T ξ , ξ, η, g) is a contact metric structure [8], which implies [26] ( Suppose that (g, T) satisfy (47b-i). By (47g), (47h), the left-hand side of the equation dual to (47a) reduces to − T ,T . Hence, by (47a) and (55) 2 , a pair (g, T) is critical for the action (6) with respect to volume-preserving adapted variations if and only if Taking a local orthonormal basis of D, where for 1 ≤ i ≤ p 2 we have E i+p/2 = T ξ E i , and using (55) 1 , we obtain as φ is symmetric in its arguments. It follows from (57) that and taking trace of (56) we obtain that div((Tr ⊥ T) ) = p(p + 5) p + 4 .
By the Divergence Theorem this cannot hold on closed M . On the other hand, if M is not closed, let T ξ ξ = 0 and for all X, Y, Z ∈ D let (T ξ X) = 0, (T X ξ) = 0, where f ∈ C ∞ (M ), and let T ξ X, Y = T ξ X, Y , and T X Y, Z = ω(X, Y, Z), where ω is any 3-form. Then (56) holds if and only if Corollary 7 can be viewed as an integrability condition for (47a). Below we give examples of T, constructed for metrics g that satisfy (50) with particular form of χ, obtaining pairs (g, T) that are critical points of (6) with respect to variations of T and g ⊥ -variations of metric. Proof. Suppose that T X Y ∈ X ⊥ for all X, Y ∈ X ⊥ . Then φ = 0, χ = 0, see definitions (46) (because T j E a , E i = − T j E i , E a = 0), (Tr ⊥ T) = 0, from equations for critical connections it follows that D is integrable and (47a) is an algebraic equation for symmetric (0,2)-tensor φ: For H = 0, we can always find φ (and then T) satisfying (58). Clearly, such φ is not unique.

Proposition 9.
Let n, p > 1 and H = 0 everywhere on M . For any g such that D is totally umbilical and D is totally geodesic and (50) holds with χ = − T , there exists a contorsion tensor T such that (T X ξ) ⊥ =T ξ X for all X ∈ X ⊥ , ξ ∈ X , and a pair (g, T) is critical for the action (6) with respect to g ⊥variations of metric and arbitrary variations of T.
we also get (Tr ⊥ T) = 0 =H and similarly, φ = 0. So, (47a) has the following form: Again, we get an algebraic equation for symmetric tensor φ, which admits many solutions.
Note that in Propostions 8 and 9 instead of condition H = 0 everywhere on M , we can assume that at those points of M , where H = 0 the metric g satisfies (58) and (59) with H = 0 (then these equations do not contain φ).
We define components of T with respect to the adapted frame on U . Let (T i E j − T j E i ) = 0 for i = j and let (T i E a ) , (T a E b ) ⊥ and (T a E i ) ⊥ be such that (47c,e,f,h) hold on U . For all (i, j) = (p, p), consider (49) evaluated on (E i , E j ) as a system of linear, non-homogeneous, first-order PDEs for {φ(E i , E j ), (i, j) = (p, p)}, assume in this system that φ(E p , , and let {φ ij , (i, j) = (p, p)} be any local solution of this system of PDEs on (a subset of) U . Let , then (47d, i) hold. By the assumption that (50) holds and the fact that (49) is a linear, non-homogeneous equation for φ, (49) evaluated on (E p , E p ) will also be satisfied. Thus, equations (47b-i) and (49) hold on (a subset of) U for T constructed above.
Note that when we consider adapted variations, we also have the equation dual (with respect to interchanging D and D) to (47a), so we can mix different assumptions from the above examples for different distributions, e.g., conditions (T i E a ) ⊥ =T a E i and T X Y ∈ X for X, Y ∈ X .

Semi-symmetric Connections
The following connections are metric compatible, see [33]. Using variations of T in this class, we obtain example with explicitly given tensor Ric D .

Definition 5. An affine connection∇ on
where U = ω is the dual vector field.
We find Euler-Lagrange equations of (4) as a particular case of (30a-h), using variations of T corresponding to semi-symmetric connections. Now we consider variations of a semi-symmetric connection only among connections also satisfying (60) for some U .
Proof. Let U t , t ∈ (− , ), be a family of compactly supported vector fields on M , and let U = U 0 andU = ∂ t U t | t=0 . Then for a fixed metric g, from (82) we obtain Separating parts with (U ) and (U ) ⊥ , we get from which (61) follow.

Remark 4.
Using computations from Lemma 6, we can show that if a semisymmetric connection∇ on (M, g, D) is critical for the action (6) with fixed g, then both D and D are integrable and totally geodesic. Indeed, let∇ be given by (60) and satisfy (47b-g) and conditions (47h), i.e., it is critical for action (6) with fixed g. We find from (85) that both D and D are integrable. Moreover, if n = p = 1 then (84) and its dual with (47b-g) yield H = 0 =H and U = 0 (i.e., the connection∇ becomes the Levi-Civita connection). If n > 2 and p > 2 we also have H = 0 =H and U = 0, in this case using also (47h). If n = 1 and p > 1 we obtain from (47d) that U ⊥ = 0 and from and Proof. By Proposition 2 and (83), we obtain Using (27a,b) give rise to (62a,b). Finally, notice that (63) is (61) for vacuum space-time.
Although generally Ric D in (8) has a long expression and is not given here, for particular case of semi-symmetric connections, due to Theorem 6, we present the mixed Ricci tensor explicitly as also S D = Tr g Ric D = S D + 2 2−n−p Z, where Ric D and S D as in Definition 3, n + p > 2 and where Θ = T − T * + T ∧ − T * ∧ and for any (1, 2)-tensor P we have P ∧ * X Y, Z = P ∧ X Z, Y = P Z X, Y = P * Z Y, X , for all X, Y ∈ X M . The following equalities (and similar formulas for Υ α,α , Υ θ,α , etc.) will be used (here S is a symmetric (0, 2)-tensor, recall Remark 1 for other notational conventions): The variations of components of Q in (18) (used in previous sections) are collected in the following three lemmas; the results for g variations are dual to g ⊥ -parts in results for g -variations.
Lemma 2. For any g -variation of metric g ∈ Riem(M, D, D) we have Proof. For any variation g t of metric and X, Y ∈ X M we have , where the first formula is obvious, the second one follows from (19) 1 , equality ∂ t T = 0 and Proof. To obtain ∂ t Θ, we compute for X, Y, Z ∈ X M : On the other hand, We shall use Proposition 3 and the fact that for Summing the 8 terms computed above and simplifying, we obtain (67). Proof of (68). We have We start from the fourth term of the 6 terms above. Then, from [26], We have We also have Now we consider other terms of ∂ t Θ, A . For the fifth term we have For the first, second and third terms we have Using (76), we have Hence, for the sixth term of ∂ t Θ, A , we have So finally we get (68). Proof of (69). We have because ∂ t T = 0. We compute 5 terms above separately: Finally, we get (69). Proof of (70). We have Let U : D × D → D be a (1, 2)-tensor, given by We compute the fifth term in ∂ t Θ,T : thus, using (1,2)-tensor F defined in (66), we can write For the first four terms of ∂ t Θ,T , see (77), we obtain: Using (76), we consider which can be simplified to the following: Hence, the sixth term in ∂ t Θ,T is: Finally, we get (70). Proof of (71). We have Hence We shall denote by (h) the fifth of the above 6 terms, and write it as sum of seven terms (h1) to (h7): We have for the term (h1) above: which can be written as = −2 div B, L + 2 B, div L . For (h2): Note that for (h3) we can assume ∇ X E a ∈ D for all X ∈ T M at a point, where we compute the formula, and hence For (h5), analogously, For (h6) term we have and (h7) term can be written as Now we compute other terms of ∂ t Θ,Ã . Recall that those 6 terms are For the first and second terms of the above ∂ t Θ,Ã we have For the third and fourth terms we have: For the sixth term, note that Finally, we get (71). Proof of (72) and (73) is straightforward. Proof of (74) and (75). The variation formulas for these terms appear in the following part of Q in (18) Then we have