Invariants for Second Type Almost Geodesic Mappings of Symmetric Affine Connection Space

: This paper presents the results concerning a space of invariants for second type almost geodesic mappings. After discussing the general formulas of invariants for mappings of symmetric affine connection spaces, based on these formulas, invariants for second type almost geodesic mappings of symmetric affine connection spaces and Riemannian spaces are obtained, as well as their mutual connection. Also, one invariant of Thomas type and two invariants of Weyl type for almost geodesic mappings of the second type were attained.


Introduction
In this research, invariants for almost geodesic mappings of the second type of a symmetric affine connection space are obtained.Invariants for second type almost geodesic mappings of a Riemannian space are obtained as a special case.
This research is based on symmetric affine connection spaces and Riemannian spaces in the sense of Eisenhart's definitions [1,2].
The covariant derivative of a tensor a α β with respect to the affine connection 0 ∇ in the direction of x γ is defined as [3][4][5] for partial derivative ∂/∂x γ denoted by comma.
From the alternation a α β|γ|δ − a α β|δ|γ , one Ricci identity is obtained (for details, see [3,5]).In this way, one curvature tensor of space A N is defined 0 The corresponding Ricci tensor 0 The alternation of the Ricci tensor is 0 After involving the abbreviation

Riemannian Space in Eisenhart's Sense
A special kind of symmetric affine connection spaces are referred to as Riemannian spaces.An N-dimensional manifold M N equipped with symmetric metric tensor ĝ, whose components are g αβ , g αβ = g βα , det g αβ ̸ = 0, is the Riemannian space R N see [1,3,5] .Because of the regularity of matrix g αβ , the metric tensor with upper indices is defined as The affine connection coefficients of space R N are the second kind of Christoffell symbols The second kind of Christoffell symbols, Γ α βγ , are symmetric by β and γ uniquely generate the torsion-free affine connection 0 ∇ g .With respect to this affine connection, one kind of covariant derivative of the tensor a α β in the direction of x γ is [3,5] The curvature tensor and the Ricci tensor of space R N are 0 The Ricci tensor 0 R g αβ is symmetric by α and β, i.e., it holds the equation 0

Almost Geodesic Mappings
In an attempt to generalize the concept of geodesics, N. S. Sinyukov defined the almost geodesic curve of a space A N as a curve ℓ = ℓ(t) which satisfies the next system of partial differential equations [3][4][5][9][10][11][12][13] where λ = dℓ/dt is tangential vector to ℓ, and a(t) and b(t) are functions of t.
A curve ℓ = ℓ(t) is an almost geodesic line of Riemannian space R N if the following system of partial differential equations is satisfied A mapping f : A N → ĀN which any geodesic line of the space A N transforms to an almost geodesic line of the space ĀN is the almost geodesic mapping of A N .
A mapping f : R N → RN which any geodesic line of the space R N transforms to an almost geodesic line of the space RN is the almost geodesic mapping of R N .
The expressions of invariant b as correspond to three types of almost geodesic mappings of space A N .These types are π 1 , π 2 , and π 3 .
After reducing the Equation ( 13) to the case of affine connections 0 ∇ g and 0 ∇g of Riemannian spaces R N and RN , the necessary and sufficient condition for a mapping f : R N → RN to be almost geodesic is for the deformation tensor P gα βγ = Γα βγ − Γ α βγ .As in the case of almost geodesic mappings of symmetric affine connection spaces, there are three types of almost geodesic mappings of Riemannian spaces as well.These three types are determined with the expressions (14) of invariant b.
The mapping f has the property of reciprocity if the affinor F α β is an invariant for this mapping and the inverse mapping f −1 is an almost geodesic mapping of the second type.The basic equations for the second type almost geodesic mapping f , which has the property of reciprocity, are Equation ( 16) together with the condition Almost geodesic mappings of symmetric affine connection space of the second type are elements of class π 2 .The subclass of almost geodesic mappings of the second type, which have the property of reciprocity, is π 2 (e).
After expressing the difference Lα βγ − L α βγ as it was obtained see [17] that the geometrical objects 0 are invariants for the mapping f : A N → ĀN whose deformation tensor is given by (18).These invariants are the basic invariants of Thomas and Weyl type for mapping f , respectively.In [19,20], two kinds of invariance of geometrical objects under mappings of nonsymmetric affine connection spaces are defined.Non-symmetricity is not of great importance in the next definition.Definition 1.Let f be a mapping between two affine connection spaces, both symmetric or nonsymmetric affine connected ones, and let U α 1 ...α p β 1 ...β q be a geometrical object of type (p, q), p, q ∈ N 0 .1.

2.
If the transformation f preserves both the value and the form of geometrical object U α 1 ...α p β 1 ...β q , then the invariance for geometrical object under the transformation f is total.

Motivation
Invariants for different geometric mappings of symmetric affine connection spaces and Riemannian spaces have been obtained [21,22].
Scalar curvature π of a 2n-dimensional Riemannian manifold is defined in [23].This scalar curvature and the corresponding scalar curvature obtained with respect to the corresponding complex metric are correlated in [23].
The perturbation of metric ( 21)→( 22), ĝ → ĝ, induces the perturbation of Christoffell symbols and components of curvature tensor induced by the FLRW metric.If g = det g µν , µ, ν = 0, 1, 2, 3, is the determinant of the perturbed metric then the corresponding Einstein-Hilbert action is for the Einstein gravitational constant κ = 2.08 × 10 −43 N −1 , scalar curvature R g obtained from the perturbed metric, and the term L M describing any matter fields from the theory.
The action S should be invariant under the variation, i.e., it would be [24,25] for variational derivative δ/δg µν .The last relation is equivalent to the equations of motion for the energy-momentum tensor T µν .
The Weyl conformal curvature tensor with respect to the perturbed metric is [3,5,21] The traces C α µνα , C α µαν , C α αµν of the Weyl conformal tensor vanish.That means that it is not possible to contract the geometrical object C π µνσ by π and some of the covariant indices, µ, ν, σ, to obtain a non-trivial invariant of the form R g µν + D µν , where D µν is a tensor of the type (0, 2).
Motivated by the trace-free Weyl conformal tensor, R. Bach proposed a quadratic action [26] which is invariant under the conformal group (the group of transformations from the space to itself that preserve angles).From the last action, the modified equations of motion are obtained.
With respect to the transformations in cosmology, and the methodology for obtaining the Einstein tensor R g µν − 1 2 R g g µν , we are motivated to obtain invariants from the transformation of the curvature tensor under second type almost geodesic mappings.Unlike in the case of the Weyl conformal tensor, the trace of one of these invariants will not be identically equal to zero.For this reason, in future work, our results will be applicable for research in cosmology analogously, as in (26), but for linear cosmological models.
In the next part, the main aims of the paper are presented.

1.
To review results about invariants for mappings of symmetric affine connection spaces obtained in [18].2.
To express the deformation tensor P α βγ of second type almost geodesic mapping f : A N → ĀN in the form for tensors ρα βγ and ρ α βγ symmetric by β and γ and obtain the corresponding basic invariants (19 and 20) for almost geodesic mappings of second type of space A N .3.
To obtain the corresponding invariants for second type almost geodesic mappings of Riemannian space R N .

Review of Basic and Derived Invariants
Let us consider a mapping f : A N → ĀN whose deformation tensor P α βγ is for tensors ρ α βγ and ρα βγ symmetric by covariant indices.The forthcoming theorem is going to be proved.Theorem 1.Let f : A N → ĀN be a mapping of symmetric affine connection space A N , whose deformation tensor is given by (18).
The geometrical objects are the basic invariants of Thomas and Weyl type for the mapping f .If ρα βγ ρδ ϵζ = ρ α βγ ρ δ ϵζ , for the geometrical objects ρ α βγ and ρα βγ used in the basic Equation ( 27), the invariant The derived invariant of Weyl type for the mapping f is the geometrical object If ρα βγ ρδ ϵζ = ρ α βγ ρ δ ϵζ , the derived invariant of Weyl type for the mapping f is The invariants for mapping f listed in this theorem are totalled.
Proof.After contracting the Equation ( 27) by α and γ, one obtains Hence, the Equation ( 27) transforms to After comparing the Equations ( 33) with (18), we obtain Hence, the basic invariants of Thomas and Weyl type for mapping [17,18] f are 0 In the case of ρα βγ ρδ ϵζ = ρ α βγ ρ δ ϵζ , the basic invariant 0 W α βγδ given by (29) reduces to 0 After contracting the difference 0 = for the corresponding tensor X βγ .If expressing X βγ from the last equality, it becomes Based on this computation, it is proved that is 0 Using simple calculus, one finds that traces Because the invariants 0 T α βγ and 0 T α βγ have the same form, the basic invariant of Thomas type for the mapping f is total.The basic invariant of Weyl type for the mapping f is obtained with respect to the functional combination of the basic invariants of Thomas type.Because this combination does not affect the form of the resulting object, the basic invariant of Weyl type for the mapping f is total.The derived invariant for the mapping f is obtained by contraction of equality 0 = 0 W α βγδ − 0 W α βγδ by α and δ.For this reason, and because the basic invariant of Weyl type for the mapping f is total, the derived invariant for the mapping f is total, too.

Corollary 1. The geometrical objects
are basic invariants for mapping f : R N → RN determined by If The derived invariant of Weyl type for the mapping f is which reduces to in the case of ρα βγ ρδ ϵζ = ρ α βγ ρ δ ϵζ .The invariants for mapping f listed in this corollary are total.

Invariants for Second Type Almost Geodesic Mappings of Space A N
The next theorem will be proved below.
Theorem 2. Let f : A N → ĀN be a second type almost geodesic mapping of a symmetric affine connection space A N .The geometrical objects for ρ α βγ = −σ γ F α β − σ β F α γ , are the basic invariants of Thomas and Weyl type for the mapping f .The geometrical object for ρ α βγ and the corresponding covariant derivatives as in the basic invariants (41) and (42), is the derived invariant for mapping f .The invariant AG 2 0 T α βγ for mapping f is total.
The invariants AG 2 0 W α βγδ and AG 2 0 W α βγδ for mapping f are valued.They are total if and only if the mapping f has the property of reciprocity.
Invariants for π 2 -Mappings of Space R N A mapping f : R N → RN determined with basic equations for 1-forms ψ β , σ β , µ β , ν β , and affinor F α β , is the second type almost geodesic mapping of Riemannian space R N .
The almost geodesic mapping f has the property of reciprocity if its inverse mapping, f −1 : RN → R N , is an almost geodesic mapping of second type and affinor F α β is an invariant for this mapping.
The necessary and sufficient condition for almost geodesic mapping f of space R N to have the property of reciprocity is given by (17).
Analogously as above, the validity of the next theorem will be confirmed.
for ρ α βγ == −σ γ F α β − σ β F α γ are the basic invariants of Thomas and Weyl type for the mapping f .The geometrical object for ρ α βγ and the corresponding covariant derivatives as in the basic invariants (45) and (46), is the derived invariant for mapping f .The invariant AG 2 0 T gα βγ for mapping f is total.
The invariants AG 2 0 W gα βγδ and AG 2 0 W gα βγδ for mapping f are valued.They are totalled if and only if the mapping f has the property of reciprocity.
Proof.Let f : R N → RN be an almost geodesic mapping of the second type.The basic equations of this mapping are given by (44).
From the first of these equations, we recognize that is For the geometrical object ρ α βγ given in this way, the following equations hold: for F = F α α and F β = F | g β .

Conclusions
In this research, we obtained different invariants for almost geodesic mappings of the second type defined on symmetric affine connection spaces and on Riemannian spaces, as well.The achieved results are as follows: The results about invariants presented in [17] were reviewed.Through this review, the general formula of invariants for mappings of symmetric affine connection spaces was accentuated.
The review of results obtained in [17] was completed with the formula of invariants with respect to mappings whose deformation tensor is expressed in the form (27). Together with this formula, the definition of two types of invariants was reviewed [20].
One invariant of Thomas type and two invariants of Weyl type were obtained through a review of results from [17,18].As the main result of this research, one invariant of Thomas type (the basic one) and two invariants of Weyl type (the basic and the derived ones) for second type almost geodesic mappings of the type π 2 were obtained.
The obtained invariants of Weyl type for second almost geodesic mappings were totalled if and only if the mapping had the property of reciprocity.Otherwise, these mappings were valued.The invariants of Thomas type for second type almost geodesic mappings were totalled.It was the last result achieved in this research.
Using the difference AG 2 W gα µνα − AG 2 W gα µνα = 0, the variation of Einstein tensor g µν R g , δE µν = Ēµν − E µν , under the second type almost geodesic mapping f : R 4 → R4 could be obtained.In this study, the transformation rules of self dual affine connections and the corresponding transformation rules of affine connection coefficients and the corresponding curvature tensors under second type almost geodesic mappings were analysed.In future research, the analysis of invariants for second type almost geodesic mappings equipped with the affine connection ∇ and the corresponding dual affine connection ∇ * defined in [27,28] are going to be studied.
by α and β, one obtains the trivial equality.On the other hand, if one contracts the equality 0 = of the form (29), by α and δ, one obtains 0 by α and any of the covariant indices.The following equalities hold 0

Theorem 3 .
Let f : R N → RN be a second type almost geodesic mapping of a Riemannian space R N .The geometrical objects AG 2 0

Theorem 4 .
The geometrical objects GA 2 W gαβγδ , together with the difference L α βγ − Γ α βγ = P α βγ , the proof of this theorem is completed.