DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 185–189

INFINITESIMAL GEODESIC DEFORMATIONS OF THE TOTALLY GEODESIC MANIFOLDS

M. L. GAVRILCHENKO, N. N. KINZERSKA

Abstract. We study infinitesimal geodesic deformations of the totally geodesic manifolds. We find conditions for which totally geodesic manifolds admit nontrivial geodesic deformation. The conditions give fundamental equations of these deformations. These equations are the main goal in this paper.

1. Let V_{m} be the Riemannian space with local coordinates (y^{1}, y^{2}, …, y^{m}), then the formulas

(1)  y^{α} = f^{α}(x^{1}, x^{2}, …, x^{n}),  Rank ∥\frac{∂f^{α}}{∂x^{i}} ∥ = n < m

define a subspace V_{n} ⊂ V_{m}.

If a_{αβ} and g_{ij} are metric tensors of V_{m} and V_{n} respectively, then [1]

g_{ij} = a_{αβ} \frac{∂y^{α}}{∂x^{i}} \frac{∂y^{β}}{∂x^{j}} .

Here and further α, β, γ, … = 1, 2, …, m; i, j, k, … = 1, 2, …, n, if not stipulated otherwise.

Let ξ^{α}(x^{1}, x^{2}, …, x^{n}) be some contravariant vector field of the space V_{m} given in points V_{n}. Formulas

(2)  y^{α} = y^{α}(x) + εξ^{α}(x)

where ε is a small real parameter, define some Ṽ_{n}, which we shall name an infinitesimal deformations of V_{n}. The vector ξ^{α} is called the field of displacements or vector of displacements. We will neglect the ε in k-th degree, k = 2, 3, … because it is very small.

Thus, we consider infinitesimal deformations of the first order.

Definition. Infinitesimal deformations of the first order are called geodesic if the geodesic lines of V_{n} are preserved under these deformations [2,3,4].

1991 Mathematics Subject Classification. 53B20, 53B25.

Key words and phrases. Geodesic deformations, infinitesimal.

The paper is in the final form and no version of it will be submitted for publication elsewhere.

The following Theorem holds [2,4]:

Theorem 1. The vector field ξ^{α}(x) defines a geodesic deformation of V_{n} if and only if it is a solution of the following system of equations

(3)  \begin{matrix}a_{αβ} y_{, i}^{α}ξ_{, jk}^{β} + a_{αβ}ξ_{, i}^{α} y_{, jk}^{β}+ \\ Γ_{γβα}(y_{, i}^{α} y_{, j}^{β} ξ_{, k}^{γ} + y_{, i}^{α} ξ_{, j}^{β} y_{, k}^{γ} + ξ_{, i}^{α} y_{, j}^{β} y_{, k}^{γ})+ \\ ξ^{γ} (\frac{∂a_{αβ}}{∂y^{γ}} y_{, i}^{α}y_{, jk}^{β} + \frac{∂Γ_{γβα}}{∂y^{γ}} y_{, i}^{α}y_{, j}^{β}y_{, k}^{γ}) = ψ_{i}g_{jk} + ψ_{j}g_{ik} , \end{matrix}

where Γ_{γβα} are the Christoffel symbols constructed for a_{αβ}, ψ_{i} ≡ ψ_{,i} is a gradient vector, the comma means covariant derivative of V_{n}.

The rather complicated system (3) gains a more convenient aspect in case m = n + 1, i.e. V_{n} is hypersurface of V_{m}.

The condition (1) allows vectors y_{, i}^{α} to be chosen as basis of V_{n}. We consider, that det ∥g_{ij}∥ ≠ 0, hence, normal η^{α}(x) to hypersurface is nonisotropic [1], i.e.

a_{αβ} y_{, i}^{α}η^{β} = 0;  a_{αβ} η^{α}η^{β} = e,  e = ±1.

The system of the vectors {y_{, i}^{α}; η^{α}} forms a basis of V_{n+1}.

Let

(4)  ξ^{α}(x) = λ^{i} y_{, i}^{α} + μ η^{α}

where

λ_{j} = g_{ij}λ^{i} = a_{αβ}ξ^{α} y_{, j}^{β} ; μ = e a_{αβ}ξ^{α} y^{β}.

If we use (4), Gauss and Weingarten derivation formulas for hypersurfaces, then the system (3) can be simplified to [2,3,4]

(5)  λ_{i,jk} = λ_{s}R_{kji}^{s} + ψ_{j}g_{ik} + ψ_{k}g_{ij} + (μΩ_{ij})_{,k} + (μΩ_{ik})_{,j} − (μΩ_{jk})_{,i} ,

where R_{ijk}^{h} is the Riemannian tensor and Ω_{ij} is the second fundamental tensor V_{n} ⊂ V_{n+1}.

The system (5) also gives necessary and sufficient conditions for geodesic deformations of hypersurface V_{n}, we will name it fundamental system.

2. Let V_{n} be a totally geodesic hypersurface. It means, that geodesics of V_{n} are geodesics of V_{n+1} at the same time [1].

The condition

Ω_{ij} = 0

is an analytical feature of the totally geodesic hypersurface and then (5) has the simple form

(6)  λ_{i,jk} = λ_{s}R_{kji}^{s} + ψ_{j}g_{ik} + ψ_{k}g_{ij} .

INFINITESIMAL GEODESIC DEFORMATIONS … 187

The system (6) does not put any conditions on the invariant μ and we have Theorem 2. The vector field of displacements

ξ^{α}(x) = λ^{i} y_{, i}^{α} + μ η^{α}

has been defined up to the normal component μ at the infinitesimal geodesic deformation of the totally geodesic hypersurface V_{n} ⊂ V_{n+1}.

Remark. If V_{n} coincides with V_{m}, (2) defines the infinitesimal geodesic transformations of V_{n}, which have been studied in [1]. In this case y^{α} = x^{α}, and μ = 0, ξ^{α} = λ^{α} in (4), hence (5) has the form (6).

Thus, the problems of the infinitesimal geodesic deformations of the totally geodesic hypersurfaces and the problems of the infinitesimal transformations of any V_{n}, which preserve the geodesics of these V_{n}, are equivalent [1, page 275].

3. The problem of the infinitesimal geodesic deformations of the totally geodesic manifolds is reduced to solving the fundamental system (6). Let’s find the conditions of integrability.

Differentiating (6) covariantly with respect to x^{l} and alternating (without division) the result on k and l, we obtain

(7) ψ,_{jl} g_{ki}−ψ,_{jk} g_{li} = λ_{s,j}R_{ikl}^{s}+λ_{s,i}R_{jkl}^{s}+λ_{s,i}R_{kij}^{s}−λ_{s,k}R_{lij}^{s}+λ_{s}R_{kij,l}^{s}−λ_{s}R_{lij,k}^{s} .

Contracting equation (7) by g^{il}, we obtain

(8)  ψ,_{jk} = \frac{1}{n − 1}(λ_{s}R_{jk,}^{s} + λ_{s,k}R_{j}^{s} + λ_{s,j}R_{k}^{s})

where R_{ij} = g_{is}R_{j}^{s} is the Ricci tensor.

If we put

(9)  λ_{s,p} = b_{sp},

then (6) and (8) can be rewritten as

(10)  b_{ij,k} = λ_{s}R_{kji}^{s} + ψ_{j}g_{ik} + ψ_{k}g_{ij} ,
(11)  ψ,_{jk} = \frac{1}{n − 1}(λ_{s}R_{jk,}^{s} + b_{sp}(R_{j}^{s}δ_{k}^{p} + R_{k}^{s}δ_{j}^{p})

where δ_{k}^{p} is the Kronecker’s symbol.

The set of the equations (9), (10), (11) is the system of the Cauchy type with respect to b_{ij}, λ_{i}, ψ_{j}.

We obtain

Theorem 3. A totally geodesic manifold V_{n} admits nontrivial infinitesimal geodesic deformations if and only if the system of the equations (9), (10), (11) has a nontrivial solution.

The common solution of this system contains no more than n(n + 2) arbitrary constants, generated by initial values of searched functions. The number of linearly independent solutions of this system does not exceed n(n+2) and solutions cannot have a functional arbitrariness [4,5].

The system of the equations (9), (10), (11) is solvable only in case of realization of integrability conditions.

They hold identically for the equations (9).

The conditions of integrability for the equations (10) are the relations (7). From (8) and (9), formula (7) is given by

(12)  b_{pq}A_{ijkl}^{pq} + λ_{s}B_{ijkl}^{s} = 0,

where

A_{ijkl}^{pq} = −δ_{j}^{q}W_{i . kl}^{p} − δ_{k}^{q}W_{ij . l}^{p} + δ_{i}^{q}W_{ijk .}^{p} + δ_{i}^{p}W _{jkl}^{q} , B_{ijkl}^{s} = −g^{sh}W_{ijkl, h}

and W_{ijk}^{h} is the Weyl tensor [1 – 6].

If we use the concept of Lie derivative with respect to λ^{s} [6], (12) gets an especially convenient and compact form

(13)  L_{λ}W_{ijkl} = (λ_{i,s} + λ_{s,i})W_{jkl}^{s} .

Let’s find the conditions of integrability of (11). Having differentiated (11) with respect to x^{l}, and alternating result on k and l, applying the Ricci identity, we obtain

(14)  b_{sp}D_{jkl}^{sp} + λ_{p}R_{sjkl, . .}^{sp} + (1 − n)ψ_{s}W_{jkl}^{s} = 0

where

(15)  D_{jkl}^{sp} = −g^{sh} (δ_{l}^{p}R_{jhk,q}^{q} + δ_{k}^{p}R_{jhl,q}^{q} + δ_{j}^{p}R_{hlk,q}^{q}) .

We rewrite the formulas (14) and (15)

(16)  L_{λ}R_{jlk,q}^{q} = (n − 1)ψ_{p}W_{jlk}^{p}

or

L_{λ}W_{jkl,q}^{q} = (n − 2)ψ_{p}W_{jlk}^{p} .

The conditions of integrability (13) and (16) of the system of the equations (9), (10), (11) are linear homogeneous algebraic equations with respect to b_{ij}, λ_{i}, ψ_{i}, with factors, which are well defined on the space V_{n}.

Their differential prolongations of any order will be the same. But the number of linearly independent among them cannot exceed n(n + 2) [4,5].

Theorem 4. The totally geodesic manifold V_{n} admits nontrivial infinitesimal geodesic deformations if and only if the system of the equations (13), (16) and its linearly independent differential prolongations have nontrivial solutions with respect to b_{ij}, λ_{i}, ψ_{i}.

The common solution of the systems of the equations (9), (10), (11) consists of no more than n(n + 2) essential parameters. We will name their amount as the degree of mobility V_{n} concerning tangent geodesic deformations, i.e. when the field of displacements is given by

ξ^{α} = λ^{i}y_{, i}^{α} .

The space V_{n} will have the maximum degree of mobility, for which (13) and (16) hold identically. It is obvious that it will hold if and only if W_{ijkl} = 0 or, in other words, for spaces of constant curvature.

Theorem 5. Only the Riemannian spaces of constant curvature have the maximum degree of mobility concerning tangent geodesic deformations, which equals n(n + 2).

References

[1] Eisenhart, L. P., Riemannian geometry, Pergamenn Press, 1925.

[2] Gavrilchenko, M. L., Geodesic deformations of Riemannian spaces, Diff. geom. and its appl., Conf. Brno, 1989, - Singapore etc., 1990, 47–53.

[3] Mikeš, J., Geodesic mappings of affine-connection and Riemannian spaces, J. Math. Sci., New York, 1996, 311–333.

[4] Sinyukov, N. S., Geodesic mappings of Riemannian spaces, Moscow, Nauka, 1979.

[5] Radulovich, Zh., Mikesh, J., Gavrilchenko, M. L., Geodesic mappings and deformations of Riemannian spaces, CID, Podgorica, 1997.

[6] Yano, K., Bochner, S., Curvature and Betti numbers, Princeton, New Jersey, 1953.

Dept. of Geometry and Topology, IMEM, Odessa state university Petra Velikogo 2, Odessa, UKRAJINE E-mail: kye@farlep.net