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

. We study two Einstein–Hilbert type actions on an almost-product metric-aﬃne manifold, considered as functionals of the contorsion tensor. The ﬁrst one is the total mixed scalar curvature of the linear connection, and the second one is based on a new type of curvature, recently introduced by B. Opozda for statistical structures. We deduce Euler–Lagrange equations of the actions and examine critical contorsion tensors associated with general and distinguished classes of connections, e.g. metric, statistical and adapted. The existence of such critical tensors depends on simple geometric properties of the almost-product structure, expressed only in terms of the Levi-Civita connection.


Introduction
Distributions on manifolds appear in various situations, e.g. as tangent bundles of foliations or kernels of differential forms. An important role in understanding geometry of distributions and foliations play linear connections and the mixed sectional curvature, i.e., sectional curvature of planes that non-trivially intersect the distribution and its orthogonal complement, see [2,7]. The mixed scalar curvature, i.e., an averaged mixed sectional curvature, is one of the simplest curvature invariants of an almost product manifold. The Euler-Lagrange equations for the total mixed scalar curvature, as functional on the space of metrics, has been studied in [1] as analog of Einstein-Hilbert action, and then in [8,9] for distributions of any dimension.
The Metric-Affine Geometry (founded by E. Cartan) generalizes Riemannian Geometry: it uses an asymmetric connection with torsion,∇, instead of the Levi-Civita connection ∇ of g, and appears in such context as homogeneous and almost Hermitian manifolds, Finsler geometry and gauge theory of gravity. The important distinguished cases are: Riemann-Cartan manifolds, where metric connections, i.e.,∇g = 0, are used, e.g. [4], and statistical manifolds [3,5], where the torsion is zero and the tensor∇g is symmetric in all its entries. The main notion of Information Geometry is that of statistical manifold, and the theory of affine hypersurfaces in R n+1 is a natural source of such manifolds. Riemann-Cartan spaces are central in gauge theory of gravity, where the torsion is represented by the spin tensor of matter.
The difference T :=∇ − ∇ is called the contorsion tensor. For the curvature tensorR X,Y = [∇ Y ,∇ X ] +∇ [X,Y ] of∇, using similar formula for the curvature tensor R of ∇, we get Let M n+p be a connected manifold with a pseudo-Riemannian metric g of index q and complementary orthogonal non-degenerate distributions D and D ⊥ (subbundles of the tangent bundle T M of ranks dim R D x = n and dim R D ⊥ x = p for every x ∈ M ) called an almost-product structure on M , see [2]. When q = 0, g is a Riemannian metric, resp. a Lorentz metric when q = 1. Let and ⊥ denote g-orthogonal projections onto D and D ⊥ , respectively. The following convention is adopted for the range of indices: The function on (M, g,∇) endowed with orthogonal complementary distributions (D, D ⊥ ), is called the mixed scalar curvature w.r.t.∇. Here (and in further parts of the paper), {E a , E i } is a local orthonormal frame on M adapted to D and D ⊥ , We will also use the notation e μ and μ = g(e μ , e μ ) when we consider elements of the orthonormal adapted frame without distinction to which distribution they belong. In particular, is the mixed scalar curvature, see [10]. Definitions (3) and (4) do not depend on the order of distributions and on the choice of a local frame. In [5], the K-sectional curvature of a symmetric (1, 2)-tensor K (on any subspace of a vector space endowed with a scalar product and a cubic form) was introduced and applied to statistical connections. It is defined for any X, Y ∈ X M by the following formula: K(X, Y ) = g([K X , K Y ] Y, X). This way, for any (1, 2)-tensor K on a pseudo-Riemannian manifold (M, g) endowed with Vol. 73 (2018) The mixed scalar curvature Page 3 of 19 23 a pair (D, D ⊥ ), we introduce the following invariant, called the mixed scalar K-curvature: If K X (X ∈ T M) is either symmetric or anti-symmetric then (5) reads Observe that the mixed scalar T-curvature (associated with contorsion tensor) can be recognized as a part ofS mix , see (3). Indeed, using (1), one can decomposeS mix of (3) into the sum: where Due to (6) and (7), we will considerS mix as a function of a (1, 2)-tensor T.
We study (1, 2)-tensors T on (M, g), which are critical for the functionals The integral is taken over M if it converges; otherwise, one integrates over arbitrarily large, relatively compact domain Ω ⊂ M . We consider arbitrary variations∇ t = ∇ + T t , and variations corresponding to distinguished classes of connections (e.g. metric and statistical), while Ω contains supports of infinitesimal variations ∂ t T t . In such cases, the Divergence Theorem states that M (div ξ) d vol g = 0 when ξ ∈ X M is supported in Ω.
In the paper, we deduce the Euler-Lagrange equations of (8) 1 and (8) 2 and examine critical contorsion tensors T (and their connections) in general and in distinguished classes. In Sect. 2, we prove (Theorems 1, 2) that T is critical for (8) 1 if and only if both distributions are totally umbilical with respect to the Levi-Civita connection and T obeys certain linear system; for statistical connections the geometrical condition is integrability of distributions instead of their umbilicity (Theorem 3). In Sect. 3, we prove (Theorem 4) that a tensor T is critical for (8) 2 if and only if it obeys certain linear system, and for adapted connections the necessary geometric conditions are integrability and minimality of distributions. These results show how the Riemannian geometry of an almost-product manifold restricts existence of linear connections critical for (8) 1 and (8) 2 . As an example, in Sect. 4 we discuss double-twisted product of metric-affine manifolds, where above conditions can be realized. Throughout

Preliminaries
We will define several geometric objects for the almost-product structure (M, D, D ⊥ , g). Let X M (resp., X D ) be the module over C ∞ (M ) of all vector fields on M (resp. all vector fields with values in D). A metric-affine space is a manifold M endowed with a metric g of certain signature and a linear connection∇. A connection∇ : X M × X M → X M on T M has the properties: A unique metric and torsion free connection on (M, g) is the Levi-Civita connection ∇, it is given by For Z ∈ D ⊥ , X, Y ∈ D, the shape operator A Z (of D w.r.t. Z) and the operator T Z are defined by . We will use the following convention for various tensors: The following formula, see [8,10]: Vol. 73 (2018) The mixed scalar curvature Page 5 of 19 23 has found many applications. Given T, define the (1,2)-tensor T * by Two partial traces of T (and similarly, of T * ) are defined by We have the following decomposition into symmetric and antisymmetric parts: (8) 1 if and only if D and its orthogonal distribution D ⊥ are both totally umbilical and T satisfies the linear system:

Arbitrary Variations Theorem 1. Let (M, g,∇) be a metric-affine manifold endowed with a nondegenerate distribution D. A tensor T is critical for action
Proof. For the terms of Q, see (7), we have From the above formula, its dual (with respect to interchanging distributions D and D ⊥ ) and (7) we obtain Since S mix does not depend on T, for one-parameter family (9) of (1, 2)-tensors. We have Since Q 1 is linear in T, to get its t-derivatives one should replace T by S in (12) 2 . Using this together with (13a)-(13b) and their dual equations, and removing integrals of divergences of compactly supported vector fields, we get Vol. 73 (2018) The mixed scalar curvature Page 7 of 19 23 where both integrals are with respect to the volume form d vol g , all sums are taken over repeated indices and factors μ are omitted. Since no further assumptions are made about S or T, all the components g(S μ e λ , e ρ ) are independent and the above formula gives rise to the following Euler-Lagrange equations: To simplify (15a)-(15h), first we consider (15a). It may yield three equations, in the following cases: From here it follows that (Tr ⊥ T + Tr ⊥ T * ) = 0. 2. a = b = c (note that this requires n > 1): From this we obtain (11b) 1 .
3. a = c = b (note that this requires n > 1): From this we obtain (11b) 2 . Note that case a = b = c = a gives no new conditions. Next, we consider (15c), which can be presented as: Then we examine (15e), which can written as Finally, we consider (15g): which can be presented as (11a) and implies (16). Equations (15b), (15d), (15f) and (15h) are dual to the ones considered above. The antisymmetric part of (17), which is the same as antisymmetric part of (18), yields (11c), while the symmetric part of (17) is the following: On the other hand, the symmetric part of (18) reads as Taking the sum of (19) and (20), we obtain (11d), while taking the difference of those equations yields Equation (21) yields that h ⊥ is proportional to g , and so h ⊥ = 1 n H ⊥ g follows. From the trace of (21) we obtain (11e). In this way, we obtain the first half, (11a)-(11e), of the Euler-Lagrange equations. The second half, (11f)-(11j), follows by interchanging the roles of D and D ⊥ . Remark 1. Solutions of (11a)-(11j) form an affine subspace in the linear space of all tensors T. Among all solutions there exists one with minimal norm, whose properties might be interesting.

Proposition 1. All tensors critical for action
Proof. First observe that T i and T a are independent, so we can consider two halves of the Euler-Lagrange equations separately. Components T ⊥ i fully determine components (T ⊥ i ) * and are restricted only for p > 1, by p scalar equations (11e) (for p = 1 (11e) yields H ⊥ = 0 and no restrictions on T i ). All components T i are fully determined by T ⊥ i according to (11c) and (11d) (antisymmetric and symmetric part of T i , respectively). Components (T i E j ) and (T * i E j ) are restricted by p 2 n scalar equations (11a) and for n > 1, additionally by n equations (11b) 1 . Note that from (11a) it follows that (T i E i ) = (T * i E i ) ; hence, (11b) 2 yields no new restrictions.
In total, we have p 3 components of T ⊥ i restricted for p > 1 by p scalar, linear equations; 2p 2 n components (T i E j ) and (T * i E j ) , which are restricted by p 2 n scalar, linear equations and for n > 1 by p 2 n+n scalar, linear equations. For the second half of the Euler-Lagrange equations we obtain the dual result (with n and p interchanged).

Proposition 2. Let n + p > 2, then a critical point of the action (8) 1 is not an extremal point (also for variations in the subspaces of tensors T corresponding to metric connections and corresponding to statistical connections).
Proof. Let T t = T + t · S. Then the only part ofS mix that is quadratic in t comes from the difference g( Let p ≥ 2, and assume that we have Hence, σ is neither positive definite, nor negative definite. This proof can be used also for variation among metric connections, because S defined above has all necessary symmetries. For variation in the subspaces of tensors T corresponding to statistical connections the (1, 2)-tensor S is symmetric in all its indices; hence, Since for n + p > 2 not every component of S is determined by Tr D S and Tr D ⊥ S, we see that again σ is neither positive definite, nor negative definite.

Variations Corresponding to Metric Connections
Let us examine the case when T corresponds to a metric connection, i.e., ∇ = ∇+T preserves the metric:∇g = 0. Then we have the following symmetry: Using (23), we obtain Considering a variation T t with S = ∂ t T t | t=0 and differentiating (23), while keeping the metric g fixed, leads to the following condition: Also, the curvature tensorR of a metric connection∇ has the same symmetries as R, its sectional curvatureK(X, Y ) is well defined and we can interpret the mixed scalar curvature as the sum ( Remark 2. Instead of using ∇, the Euler-Lagrange equations (25a)-(25h) can be presented in terms of extrinsic geometry of the metric connection∇ = ∇+T. For example, the second fundamental formh ⊥ of D w.r.t.∇ is given bȳ

Variations Corresponding to Statistical Connections
Let us examine the case when T corresponds to a statistical connection, i.e., ∇ = ∇+T is torsionless, with symmetric tensor∇g. Then we have the following symmetries:  Proof. Note that Tr ⊥ T * = Tr ⊥ T and Tr T * = Tr T. Substituting (26) into (14), we find that the Euler-Lagrange equations consist of the system and the equations dual to the above (with interchanged roles of distributions D and D ⊥ ), which we do not write here. From (28a) with a = b = c it follows that (Tr ⊥ T) = 0. For n > 1, (28a) with a = b = c yields additionally: H = 0 = (Tr ⊥ T) . From (28b), we obtain but from the symmetry T i E j = T j E i it follows that T = 0; hence, (T i E j ) = 0. From the dual equation we obtain that also D must be integrable. Finally, from (28c) we get δ ab g(Tr ⊥ T, E i ) − 2 g(T i E b , E a ) + g(T i E b , E a ) = 0, and furthermore, hence, 2(T a E b ) ⊥ = δ ab (Tr ⊥ T) ⊥ . Equation dual to (29) yields (T a E b ) ⊥ = 0; thus we obtain (Tr ⊥ T) ⊥ = 0. The Weyl-Cartan connections∇, i.e., Tr(∇ X g) = 0 (X ∈ X M ), have been classified in [4]. Proof. In our case, we have where T and T * are defined in Sect. 2.2. From the above we immediately see that (11b) and the dual ones are satisfied, regardless of the dimensions of distributions. Also we obtain thus, (11a) is satisfied. Equations (11c), (11e) and (11d) are restrictions on (otherwise free) components: T i , Tr ⊥ T − Tr ⊥ T * and Tr ⊥ T + Tr ⊥ T * . Finally, h ⊥ = 1 n H ⊥ g is a geometric condition in terms of ∇.
We do not consider variations ofS mix on the space of tensors T corresponding to adapted connections because (8) 1 is identically zero on this subspace, since for every such tensor we haveS mix (T) = 0.