Conformal Equitorsion and Concircular Transformations in a Generalized Riemannian Space

In the beginning, the basic facts about a conformal transformations are exposed and then equitorsion conformal transformations are defined. For every five independent curvature tensors in Generalized Riemannian space, the above cited transformations are investigated and corresponding invariants-5 concircular tensors of concircular transformations are found.


Introduction
In the sense of Eisenhart's definition [1], a generalized Riemannian space (GR N ) is a differentiable N-dimensional manifold that is endowed with basic non-symmetric tensor (g ij = g ji ), where detg ij = 0.
The symmetric part of g ij is noted with g ij and antisymmetric one with g ij V . The lowering and rising of indices in GR N is defined by g ij and g ij , respectively, where g ij g ik = δ k j (detg ij = 0). The Christoffel symbols in GR N are given in the next manner: a)Γ i.jk = 1 2 (g ji,k − g jk,i + g ik,j ), b)Γ i jk = g ip Γ p.jk = 1 2 g ip (g jp,k − g jk,p + g pk,i ), where, e.g., g ij,k = ∂g ij /∂x k . (1) Because of non-symmetry of the affine connection coefficients Γ i jk by indices j and k, there are four kinds of covariant differentiation in the space GR N . Namely, for a tensor a i j , these covariant derivatives are defined as: Yano in [2] investigates a conformal and concircular transformations in the R N . In that case, of course, he considers one that is Riemannian curvature tensor. De and Mandal in [3] studied concircular curvature tensors as important tensors from the differential geometric point of view. In [4][5][6][7][8][9][10][11], Mikeš et al. have studied special kinds ot transformations in Riemannian space.
In [18], another combination of five independent curvature tensors is obtained, and they are denoted by K 1 . . . K

.
For five independent tensors K θ in [19], the invariants Z θ were found, which are different from the invariantsZ θ in the present paper (see Remark 3.1, at the end). Compare e.g.,Z 1 from the present paper and Investigation of various kinds of mappings in the settings of generalized Riemannian spaces is an active research topic, numerous results were obtained in the recent years; see, for instance [20][21][22]. Very recently, conformal and concircular diffeomorphisms of generalized Riemannian spaces have been studied by M. Z Petrović, M. S. Stanković and P. Peška [23].

Equitorsion Conformal Transformation in Generalized Riemannian Space
Consider a special transformation of the objects in GR N .

Definition 1.
Conformal transformation is that one under which the basic tensor is changed according to the law where ρ(x) = ρ(x 1 , . . . , x N ) is some differentiable function of coordinates in GR N .
We see that g andḡ are considered in the common system of coordinates. The same is valid for the other geometric objects.
Because the inverse matrix for (g ij ) is the matrix (g ij ), we get and, based on (1), (6), (8), where Denote From (9), it is obtained: for the symmetric part of the connection and for the torsion tensor (double skewsymmetric part of the connection) Definition 2. An equitorsion conformal transformation of the connection in GR N is that conformal transformation (3) on the base of which the torsion is not changed, i.e., From (13), we conclude that Theorem 1. Necessary and sufficient condition for a conformal transformation of the connection to be equitorsion is

The First Curvature Tensor
The 1 st from the cited curvature tensors in GR N is [12,13] Based on (15), (9), we obtainΓ If by the transformation of the connection Γ intoΓ we write we can consider how e.g., some curvature tensors from the above mentioned independent ones are transformed.
With respect to (18), for R 1 , one obtains and substituting P from (18b): where | 1 m denotes covariant derivative of the first kind on x m . Because the 2nd addend on the right side in (20) is 0. Introducing the notation we obtain and, forR Furthermore, from whereR and putting in order: where A i jmn is given in (24). We are using the next definition from [2] Definition 3. If a conformal transformation in a Riemannian space R N : transforms every geodesic circle into geodesic circle, the function ρ(x) satisfies the partial differential equation where Such a transformation is called a concircular transformation in R N , and concircular geometry is geometry that treats the concircular transformations and the spaces that allow such kinds of transformations.
In the GR N , we consider transformations where, based on (22), ρ i| and such a transformation we name a concircular transformation of the first kind in GR N . We have to find the function Φ 1 . Substituting ρ 1 from (30) into (26), we get: If we effect the contraction with i = n, it follows that where R 1 jm = R 1 i jmi , and so on, and we get: By multiplying the corresponding sides of previous equation and the equation A i jmi g jm = (T i mi ρ j − T j.mi ρ i )g jm = 0, and we get wherefrom it follows that Substituting Φ 1 into (31), we get and from hereR Taking into consideration that with respect to (24) and (35) where T i mn =T i mn (for the first addend) and g ip g qj =ḡ ipḡ qj (for the third addend). By substituting from (36) into (34) and because of we obtainR In that manner, we conclude that the following theorem is valid: is an invariant in the space GR N , by an equitorsion concircular transformation i.e., according to (38): where e.g., g j = (ln g) ,j = ∂(ln g) ∂x j andZ 1 i jmn is given by (39).
The tensorZ 1 i jmn is an equitorsion concircular tensor of the first kind in GR N .

The Second Curvature Tensor
The tensor R 2 in GR N is [12,17] and, forR 2 i jmn , by virtue of (18), it follows that Substituting from (18) into the previous equation and arranging, one obtains The term in the 1st bracket on the right side is 0 because of If we introduce the denotation we have where A i jmn is given (24). Furthermore, we use the concircular transformation for R By substitution of ρ 2 ij into (46), by procedure as for R 1 , we obtain and at the end: where A i jmn is given in (24). Thus, we conclude that the next theorem is valid.
is an invariant in GR N with respect to an equitorsion concircular transformation, i.e., in force is The tensorZ 2 is an equitorsion concircular tensor of the 2nd kind at GR N and e.g., g j = (ln g) = ∂(ln g) ∂x j .

The Third Curvature Tensor
The tensor R 3 in GR N [12,14,17] is where T i pj is torsion tensor in local coordinates. For R 3 i jmn on the base of (18), it is obtained where we take into consideration that P i jp is symmetric, with respect to (18). By substituting from (18) into the previous equation and arranging, one obtains where D i jmn = T i jm ρ n + T i nj ρ m + g ps g mn T i jp ρ s .
From (55), it is obtained that Consider, further, the concircular transformation for the tensor R 3 i jmn in the following manner. Taking we obtain from (56)R Putting i = n, we getR and contracting with ρ 2ḡjm = g jm on the left and the right sides correspondingly in (59), we get By the further procedure as in the case of R 1 , we obtain Consider, further, the tensor D i jmn . By virtue of (35), one gets where the equitorsion is taken into consideration. Substituting from (62), (63) into (58), it follows that from where we conclude that the next theorem is valid.
(T i jm g n + T i nj g m + g ps g mn T i jp g s ) is an invariant in GR N with respect to an equitorsion concircular transformation, i.e., it is The tensorZ 3 is an equitorsion concircular tensor of the 3rd kind at GR N .

The Fourth Curvature Tensor
For the tensor R 4 in GR N , we have [13,14,17] where T i pj is torsion tensor in local coordinates. For R 4 i jmn on the base of (18), it is obtained From (53), (68), it follows that where D i jmn is given in (55). For the concircular transformation for the tensor R 4 i jmn , we put and, by the same procedure as in the previous case, the next theorem is obtained.
(T i jm g n + T i nj g m + g ps g mn T i jp g s ) is an invariant in GR N with respect to an equitorsion concircular transformation, i.e., in force is The tensorZ 4 is an equitorsion concircular tensor of the 4th kind at GR N .

The Fifth Curvature Tensor
Finally, consider the 5th curvature tensor R ). We have according to [12,17] which can be written in the form [17]: where P i jk is given in (18). With substitution of P from (18) into (73), one obtains Using (23) and (44) and introducing the denotation Let us apply a concircular transformation for the tensor R 5 i jmn . By virtue of (75), we put into (76) and we getR 5 i jmn = R 5 i jmn + 2Φ 5 (δ i m g jn − δ i n g jm ) − ρ p ρ p (δ i m g jn − δ i n g jm ).