MATEMATIQKI VESNIK 49 (1997), 27–33

UDK 514.764.3

originalni nauqni rad research paper

ON GEODESIC MAPPINGS OF GENERAL AFFINE CONNEXION SPACES AND OF GENERALIZED RIEMANNIAN SPACES

Svetislav Minčić and Mića Stanković

Abstract. In the present paper we define a geodesic mapping of two nonsymmetrical affine connexion spaces and obtain necessary and sufficient conditions that a mapping of two such spaces be geodesic (§1). Particularly we study a geodesic mapping of two generalized Riemannian spaces (§2). Finally, we generalize the notion of Thomas’s projective parameters as an invariant of geodesic mappings (§3).

1. Geodesic mappings of general affine connexion spaces

Consider two N-dimensional spaces of nonsymmetrical affine connexion: GA_{N} and G\overline{A}_{N}. So, if connexion coefficients of these spaces are respectively L_{jk}^{i} and

\overline{L}_{jk}^{i}, we suppose that in general the symmetry with respect to indices j, k is not valid.

One says that reciprocal one-valued mapping f: GA_{N} → G\overline{A}_{N} is geodesic, if geodesics of the space GA_{N} pass to geodesics of the space G\overline{A}_{N}. We can consider these spaces together with this mapping system of local coordinates, i.e. for f: M ↦ \overline{M} we have M(x^{1},…, x^{N}) ≡ M(x) and \overline{M}(x^{1},…, x^{N}) ≡ \overline{M}(x), where M ∈ GA_{N}, \overline{M} ∈ G\overline{A}_{N}. In the corresponding points M(x) and \overline{M}(x) we can put

\overline{L}_{jk}^{i}(x) = L_{jk}^{i}(x) + P_{jk}^{i}(x), (i, j, k = 1,…, N),  (1.1)

where P_{jk}^{i}(x) is the deformation tensor of the connexion L of GA_{N} according to the mapping f: GA_{N} → G\overline{A}_{N}.

The curve

l: x^{i} = x^{i}(t)  (1.2)

is geodesic of GA_{N} if and only if for λ^{i} = dx^{i}/dt it is:

\frac{dλ^{i}}{dt} + L_{pq}^{i}λ^{p}λ^{q} = ρ(t)λ^{i}(t),  (1.3)

where ρ(t) is an invariant.

AMS Subject Classification: 53 B 05

Keywords and phrases: Geodesic mapping, general affine connexion space, generalized Riemannian space, Thomas’s parameters

Communicated at the 11. Yugoslav Geometrical Seminar, Divčibare 1996.

Supported by Grant 04M03D of RFNS through Math. Inst. SANU.

If f: l → \overline{l}, then by the mapping f, coordinates \overline{x}^{i} ≡ x^{i} and it is \overline{λ}^{i} = d\overline{x}^{i}/dt = λ^{i}, and \overline{l} is geodesic in G\overline{A}_{N}, too, so we get

\frac{dλ^{i}}{dt} + \overline{L}_{pq}^{i}λ^{p}λ^{q} = \overline{ρ}(t)λ^{i}(t)  (1.\overline{3})

Subtracting (1.\overline{3}) and (1.3), we obtain

(\overline{L}_{pq}^{i} − L_{pq}^{i})λ^{p}λ^{q} = (\overline{ρ}(t) − ρ(t))λ^{i}(t),

and, because of (1.1):

P_{pq}^{i}λ^{p}λ^{q} = 2ψ(t)λ^{i}(t).  (1.4)

Denoting by P i ; \underline{jk}, P i ; jk_{∨} the symmetric and antisymmetric part of P_{jk}^{i} respectively,

we get

P_{jk}^{i} = P i ; \underline{jk} + P i ; jk_{∨},  (1.5)

and (1.4) reduces to

P i ; \underline{pq}λ^{p}λ^{q} = 2ψ(t)λ^{i}(t).  (1.4′)

As in the case of a symmetric connexion (see e.g. [4]) one concludes that ψ(t) = ψ_{p}(x^{1}(t),…, x^{N}(t))λ^{p}(t), and from (1.4′):

P i ; \underline{pq}λ^{p}λ^{q} = 2ψ_{p}λ^{p}λ^{i} = ψ_{p}λ^{p}λ^{q}δ_{q}^{i} + ψ_{q}λ^{q}λ^{p}δ_{p}^{i} = (ψ_{p}δ_{q}^{i} + ψ_{q}δ_{p}^{i})λ^{p}λ^{q},

wherefrom

P i ; \underline{jk} = δ_{j}^{i}ψ_{k} + δ_{k}^{i}ψ_{j}.  (1.6a)

Denoting also

P i ; jk_{∨} = ξ_{jk}^{i} = −ξ_{kj}^{i},  (1.6b)

substituting in (1.1) we obtain

\overline{L}_{jk}^{i} = L_{jk}^{i} + δ_{j}^{i}ψ_{k} + δ_{k}^{i}ψ_{j} + ξ_{jk}^{i},  (1.7)

and the deformation tensor is

P_{jk}^{i}(x) = δ_{j}^{i}ψ_{k}(x) + δ_{k}^{i}ψ_{j}(x) + ξ_{jk}^{i}(x).  (1.8)

So, the condition (1.8) is necessary that the mapping f be geodesic. It is easy to prove that this condition is sufficient, too, and we have

Theorem 1.1. A necessary and sufficient condition that the mapping f: GA_{N} → G\overline{A}_{N} be geodesic is that the deformation tensor P_{jk}^{i} from (1.1) according to the mapping f has the form (1.8), where ψ_{j}(x^{1},…, x^{N}) is a covariant vector, and ξ_{jk}^{i}(x^{1},…, x^{N}) an antisymmetric tensor.

For k = i, we obtain from (1.6a) that P i ; \underline{ji} = δ_{j}^{i}ψ_{i}+δ_{i}^{i}ψ_{j} = ψ_{j} +Nψ_{j}, wherefrom

ψ_{i} = \frac{1}{N + 1}P p ; \underline{ip},  (1.9)

which, by substitution in (1.7), gives

\overline{L}_{jk}^{i} = L_{jk}^{i} + \frac{1}{N + 1}(δ_{j}^{i}P p ; \underline{kp}(x) + δ_{k}^{i}P p ; \underline{jp}(x)) + P i ; jk_{∨}(x),  (1.7′)

where P_{jk}^{i}(x) is the deformation tensor.

On the base of the facts given above, we get

Theorem 1.2 Let a space GA_{N} be given, i.e. on a differentiable manifold M_{N} let nonsymmetric connexion coefficients L_{jk}^{i}(x) be defined. If on M_{N} a tensor

P_{jk}^{i}(x) is given, too and we determine \overline{L}_{jk}^{i}(x) according to (1.7′), then on M_{N} a

space G\overline{A}_{N} is defined, with connexion coefficients \overline{L}_{jk}^{i}(x), so that GA_{N} and G\overline{A}_{N} have common geodesics. We obtain the same result (on the base of (1.7)) choosing a vector ψ_{i}(x) and antisymmetric tensor ξ_{jk}^{i}(x) = P i ; jk_{∨}(x).

A question arises itself: Is there a geodesic mapping of a space GA_{N} with a nonsymmetric affine connexion onto a space \overline{A}_{N} with a symmetric affine connexion? It is easy to see that the next theorem is valid.

Theorem 1.3 A necessary and sufficient condition that a mapping f : GA_{N} → \overline{A}_{N} of a nonsymmetric affine connexion space GA_{N} onto a symmetric affine connexion space \overline{A}_{N} be geodesic, is that

P i ; jk_{∨}(x) = −Li ; jk_{∨}(x),  (1.10)

where P i ; jk_{∨}(x), Li ; jk_{∨}(x), are antisymmetric parts of the deformation tensor and con-

nexion coefficients of the GA_{N} respectively.

Remark. It is easy to check that a set of geodesic mappings of a space GA_{N} forms a group.

2. Geodesic mappings of generalized Riemannian spaces

Generalized Riemannian space GR_{N} in the sense of Eisenhart’s definition [1] is a differentiable N-dimensional manifold, equipped with nonsymmetric basic tensor g_{ij}. Connexion coefficients are generalized Cristoffel’s symbols of the second kind Γ_{jk}^{i}, where

Γ_{i.jk} = \frac{1}{2}(g_{ji,k} − g_{jk,i} + g_{ik,j}), Γ_{jk}^{i} = g\underline{ip}Γ_{p.jk},  (2.1)

and g\underline{ip}g\underline{ip} = δ_{j}^{i}.

Generally, it is Γ_{jk}^{i} ≠ Γ_{kj}^{i}. Based on this, everything that we exposed for GA_{N}, is valid for GR_{N}, too. We will expose some specifics.

In a space of nonsymmetric affine connexion (and in a generalized Riemannian space) one can define four kinds of covariant derivative [2,3]. For example, for a tensor a_{j}^{i} in GA_{N} we have

\begin{matrix}a_{j |_{1}m}^{i} = a_{j,m}^{i} + L_{pm}^{i}a_{j}^{p} − L_{jm}^{p}a_{p}^{i}, \\ a_{j |_{2}m}^{i} = a_{j,m}^{i} + L_{mp}^{i}a_{j}^{p} − L_{mj}^{p}a_{p}^{i}, \\ a_{j |_{3}m}^{i} = a_{j,m}^{i} + L_{pm}^{i}a_{j}^{p} − L_{mj}^{p}a_{p}^{i}, \\ a_{j |_{4}m}^{i} = a_{j,m}^{i} + L_{mp}^{i}a_{j}^{p} − L_{jm}^{p}a_{p}^{i}. \end{matrix}  (2.2)

Denote by |_{θ}, \underline{|}_{θ} a covariant derivative of the kind θ in GR_{N} and G\overline{R}_{N} respec-

tively. We define a geodesic mapping of GR_{N} in the similar manner as of GA_{N} (§1).

Suppose that a geodesic mapping f: GR_{N} → G\overline{R}_{N} is given. Then, with respect to (1.7), one obtains

\overline{Γ}_{jk}^{i} = Γ_{jk}^{i} + δ_{j}^{i}ψ_{k} + δ_{k}^{i}ψ_{j} + P i ; jk_{∨}.  (2.3)

and

\overline{g}_{ij |_{\overline{1}}k} = \overline{g}_{ij,k} − \overline{Γ}_{ik}^{p}\overline{g}_{pj} − \overline{Γ}_{jk}^{p}\overline{g}_{ip} =_{(2.3)}

= \overline{g}_{ij,k} − (Γ_{ik}^{p} + δ_{i}^{p}ψ_{k} + δ_{k}^{p}ψ_{i} + P p ; ik_{∨})\overline{g}_{pj} − (Γ_{jk}^{p} + δ_{j}^{p}ψ_{k} + δ_{k}^{p}ψ_{j} + P p ; jk_{∨})\overline{g}_{ip}

= (\overline{g}_{ij,k} − Γ_{ik}^{p}\overline{g}_{pj} − Γ_{jk}^{p}\overline{g}_{ip}) − \overline{g}_{ij}ψ_{k} − \overline{g}_{kj}ψ_{i} − \overline{g}_{ij}ψ_{k} − \overline{g}_{ik}ψ_{j} − P p ; ik_{∨} \overline{g}_{pj} − P p ; jk_{∨} \overline{g}_{ip}.

(2.4)

Because of \overline{g}_{ij |_{\overline{θ}}k} = \overline{g}_{\underline{ij} |_{\overline{θ}}k} + \overline{g}_{ij_{∨} |_{\overline{θ}}k} and \overline{g}_{\underline{ij} |_{\overline{θ}}k} = 0 (θ = 1, 2), and in the parentheses

on the right side of (2.4) we have \overline{g}_{ij |_{1}k}, this equation gives

\overline{g}_{ij |_{1}k} − \overline{g}_{ij_{∨} |_{\overline{1}}k} = 2\overline{g}_{ij}ψ_{k} + ψ_{i}\overline{g}_{kj} + ψ_{j}\overline{g}_{ik} + \overline{g}_{ip}P p ; jk_{∨} + \overline{g}_{pj}P p ; ik_{∨}.  (2.5a)

Starting from \overline{g}_{ij |_{\overline{2}}k}, we obtain

\overline{g}_{ij |_{2}k} − \overline{g}_{ij_{∨} |_{\overline{2}}k} = 2\overline{g}_{ij}ψ_{k} + ψ_{i}\overline{g}_{kj} + ψ_{j}\overline{g}_{ik} + \overline{g}_{ip}P p ; kj_{∨} + \overline{g}_{pj}P p ; ki_{∨},  (2.5b)

With respect to the Theorem 1.1., the condition (2.3) is necessary and sufficient that the mapping f: GR_{N} → G\overline{R}_{N} be geodesic, so the conditions (2.5a,b) are necessary for this. Let us prove that these conditions are sufficient too. Start from (2.5b). Denoting the left and right side in (2.5b) by L and R respectively, we have

L ≡ \overline{g}_{ij |_{2}k} − \overline{g}_{ij_{∨} |_{\overline{2}}k} = \overline{g}_{ij |_{2}k} − \overline{g}_{ij |_{\overline{2}}k}
= \overline{g}_{ij,k} − Γ_{ki}^{p}\overline{g}_{pj} − Γ_{kj}^{p}\overline{g}_{ip} − \overline{g}_{ij,k} − \overline{Γ}_{ki}^{p}\overline{g}_{pj} − \overline{Γ}_{kj}^{p}\overline{g}_{ip}
= (\overline{Γ}_{ki}^{p} − Γ_{ki}^{p})\overline{g}_{pj} + (\overline{Γ}_{kj}^{p} − Γ_{kj}^{p})\overline{g}_{ip},

R ≡ ψ_{i}\overline{g}_{kj} + ψ_{k}\overline{g}_{ij} + \overline{g}_{pj}P p ; ki_{∨} + ψ_{j}\overline{g}_{ik} + ψ_{k}\overline{g}_{ij} + \overline{g}_{ip}P p ; kj_{∨}

= (ψ_{i}δ_{k}^{p} + ψ_{k}δ_{i}^{p} + P p ; ki_{∨})\overline{g}_{pj} + (ψ_{j}δ_{k}^{p} + ψ_{k}δ_{j}^{p} + P p ; kj_{∨})\overline{g}_{ip}.

The equation (2.5b), i.e. L = R is satisfied for

\overline{Γ}_{kj}^{p} − Γ_{kj}^{p} = ψ_{j}δ_{k}^{p} + ψ_{k}δ_{j}^{p} + P p ; kj_{∨},

and this is the equation (1.7). Starting from (2.5a), one obtains the same. For, starting from one of the equations (2.5a,b), it follows (1.7), and from here the other of these equations follows, so we conclude that they are equivalent.

By virtue of the facts exposed above, we have

Theorem 2.1 a) A mapping f: GR_{N} → G\overline{R}_{N} is geodesic if and only together with the mapping f the system of local coordinates and the second kind Cristoffel symbols of these spaces satisfy (2.3).

b) If the mapping f is geodesic, then the equations (2.5a, b) are satisfied. Conversely, if one of these equations is satisfied, this mapping is geodesic, and the other is satisfied too.

Corollary. If the mapping f: GR_{N} → G\overline{R}_{N} is geodesic, then the basic tensor \overline{g}_{ij} of the space G\overline{R}_{N} satisfies the relation

\overline{g}_{ij |_{1}k} + \overline{g}_{ij |_{2}k} − \overline{g}_{ij_{∨} |_{\overline{1}}k} − \overline{g}_{ij_{∨} |_{\overline{2}}k} = 2ψ_{i}\overline{g}_{kj} + 2ψ_{j}\overline{g}_{ik} + 4ψ_{k}\overline{g}_{ij},  (2.6)

where ψ_{i} is a covariant vector.

This relation is obtained by adding (2.5a,b).

Further, we have

Theorem 2.2 By a geodesic mapping f: GR_{N} → G\overline{G}_{N} the vector ψ_{i} is given in the form

ψ_{i} = \frac{1}{N + 1} (ln \sqrt{|\frac{\overline{\underline{g}}}{\underline{g}} |})_{,i} ,  (2.7)

where \underline{g} = det(g\underline{ij}), \underline{\overline{g}} = det(\overline{g}\underline{ij}) and the comma denotes a partial derivative.

Proof. With respect to (2.1) one gets

Γ_{i.jk} + Γ_{j.ik} = g_{\underline{ij},k}, Γ_{i.jk} + Γ_{k.ji} = g_{\underline{ik},j}.  (2.8a, b)

On the other hand, as in the case of Riemannian space, we have:

\underline{g}_{,i} = \underline{g}g\underline{jk}g_{\underline{jk},i} = \underline{g}g\underline{jk}(Γ_{j.ki} + Γ_{k.ji}) = \underline{g}(Γ_{ki}^{k} + Γ_{ji}^{j}) = 2\underline{g}Γ_{pi}^{p},

where we used (2.8a). Using (2.8b) we obtain \underline{g}_{,i} = 2\underline{g}Γ_{ip}^{p}. From these equations we have

Γ_{pi}^{p} = Γ_{ip}^{p} = (ln \sqrt{|\underline{g}|})_{,i}.  (2.9)

Analogous equation is valid for \overline{Γ}, too, and from (1.9) one obtains

ψ_{i} = \frac{1}{N + 1}(\overline{Γ}_{ip}^{p} − Γ_{ip}^{p}) = \frac{1}{N + 1}[(ln \sqrt{|\underline{\overline{g}}|})_{,i} − (ln \sqrt{|\underline{g}|})_{,i}],

i.e. (2.7) is in effect.

Remark. From (2.9) we see that in GR_{N}, Γ_{pi}^{p} is the gradient and that

Γp ; pi_{∨} = 0.  (2.10)

From (2.7) it follows that at the mapping f: GR_{N} → G\overline{R}_{N} the corresponding vector ψ_{i} is gradient, too.

3. Generalized Thomas’s projective parameters

Putting P into (1.7′) in accordance with (1.1) we get

\overline{L}_{jk}^{i} − \overline{L}i ; jk_{∨} − \frac{1}{N + 1}(δ_{j}^{i}\overline{L}p ; \underline{kp} + δ_{k}^{i}\overline{L}p ; \underline{jp})

= L_{jk}^{i} − Li ; jk_{∨} − \frac{1}{N + 1}(δ_{j}^{i}Lp ; \underline{kp} + δ_{k}^{i}Lp ; \underline{jp}),

Denoting

T_{jk}^{i} = Li ; \underline{jk} − \frac{1}{N + 1}(δ_{j}^{i}Lp ; \underline{kp} + δ_{k}^{i}Lp ; \underline{jp}) = T_{kj}^{i},  (3.1)

we see that

\overline{T} _{jk}^{i} = T_{jk}^{i}.  (3.2)

The magnitudes T_{jk}^{i} we call generalized Thomas’s projective parameters at the mapping f: GA_{N} → G\overline{A}_{N}. Accordingly, these magnitudes are invariant at a geodesic mapping. Starting from (3.1) and (3.2), one obtains (1.7′), and we conclude that the next theorem is valid.

Theorem 3.1 A necessary and sufficient condition that a mapping f: GA_{N} → G\overline{A}_{N} be geodesic is that the generalized Thomas’s projective parameters are invariant.

Using the transformation law for connexion coefficients L_{jk}^{i}, from (3.1) we obtain the transformation law for T_{jk}^{i}(x) passing from coordinates x^{i} to coordinates x^{i′} in GA_{N}:

T_{j′k′}^{i′}(x′) = T_{jk}^{i}(x)x_{i}^{i′}x_{j′}^{j}x_{k′}^{k} − \frac{1}{N + 1}x_{i}^{i′} [x_{j′}^{i}(ln Δ)_{,k′} + x_{k′}^{i}(ln Δ)_{,j′}] + x_{i}^{i′}x_{j′k′}^{i},

(3.3)

where Δ = det(x_{i′}^{i}), x_{i′}^{i} = ∂x^{i}/∂x^{i′}, x_{j′k′}^{i} = ∂^{2}x^{i}/∂x^{j′}∂x^{k′} and so on.

Let in GA_{N} local coordinates are x^{i}, in G\overline{A}_{N}, x^{i′}, and f: GA_{N} → G\overline{A}_{N}. Passing from x^{i′} in G\overline{A}_{N} to x^{i}, we obtain an inverse relation with respect to (3.3), and (3.2) gives

T_{jk}^{i}(x) = \overline{T} _{jk}^{i}(x) = \overline{T} _{j′k′}^{i′}(x′)x_{i′}^{i}x_{j}^{j′}x_{k}^{k′} − \frac{1}{N + 1}x_{i′}^{i} [x_{j}^{i′}(ln Δ′_{,k} + x_{k}^{i′}(ln Δ′)_{,j}]+x_{i′}^{i}x_{jk}^{i′},

(3.4)

where Δ′ = det(x_{i}^{i′}).

Writing (3.4) in the form

T_{jk}^{i}(x)x_{i}^{i′} = \overline{T} _{j′k′}^{i′}(x′)x_{j}^{j′}x_{k}^{k′} − \frac{1}{N + 1} [x_{j}^{i′}(ln Δ′)_{,k} + x_{k}^{i′}(ln Δ′)_{,j}] + x_{jk}^{i′}, (3.4′)

we see that the following is valid.

Theorem 3.2 A space GA_{N}, with connexion coefficients L_{jk}^{i}(x) in local coordinates x^{i}, allows a geodesic mapping f on a space G\overline{A}_{N} with connexion coefficients \overline{L}_{j′k′}^{i′}(x′) in local coordinates x^{i′}, if and only if there exist functions

x^{i′} = x^{i′}(x^{1},…, x^{N}), (i′ = 1′,…, N′; det(x_{i}^{i′}) ≠ 0)  (3.5)

of the class C^{r} (r > 2), satisfying the equation (3.4′), and by which the mapping f: GA_{N} → G\overline{A}_{N} is realized.

Remark. The equations (3.4′) form a system of second order partial differential (nonlinear) equations with respect to unknown functions x^{i′} (3.5). But, practical solving of this problem is very difficult in general.

REFERENCES

[1] L. P. Eisenhart, Generalized Riemannian spaces I, Proc. Nat. Acad. Sci. USA 37 (1951), 311–315.

[2] S. M. Minčić, Ricci identities in the space of non-symmetric affine connexion, Matematički vesnik 25, 2 (1973), 161–172.

[3] S. M. Minčić, New commutation formulas in the non-symmetric affine connexion space, Publ. Inst. Math. (Beograd) (N.S) 36 (1977), 189–199.

[4] N. S. Sin�kov, ������������� ����������� ���������� �����������, Moskva, �Nauka�, Gl. red. fiz.-mat. lit., 1987.

(received 27.12.1996.)

University of Niš, Faculty of Philosophy, Department of Mathematics, Ćirila i Metodija 2, 18000 Niš, Yugoslavia e-mail: smica@archimed.filfak.ni.ac.yu