MATHEMATICS

V. G. LEMLEIN

PROJECTIVE AND PROJECTIVE-METRIC TRANSPORTS IN MANIFOLDS WITH AFFINE CONNECTION AND IN RIEMANNIAN SPACES

(Presented by Academician P. S. Aleksandrov on 2 III 1960)

1. Let an object ((a_i)) be given on a differentiable manifold ({V^n}), whose components, under a transformation (x^{i'}=x^{i'}(x^i)) of local coordinate systems of the manifold, transform according to the law

[
a_{i'}=\frac{\partial x^i}{\partial x^{i'}}a_i-\frac{1}{(n+1)}\frac{\partial \ln \det |\partial x^{r'}/\partial x^r|}{\partial x^{i'}}.
]

With each vector ((\xi^i)) of the tangent centro-affine space ({A^n}) let us associate the object

[
u^i=\frac{\xi^i}{-a_l\xi^l+1}
\tag{1}
]

from the local centro-projective space ({P^n}) ((^1)), and with each covector ((\xi_i)) from the space ({B^n}), dual to ({A^n}), the object

[
u_i=\xi_i+a_i
\tag{2}
]

from the space ({Q^n}), dual to ({P^n}).

The correspondences thus defined are one-to-one, since formulas (1) and (2) can be rewritten in the form

[
\xi^i=\frac{u^i}{a_lu^l+1},
\tag{3}
]

[
\xi_i=u_i-a_i
\tag{4}
]

and do not depend on the choice of coordinate system on ({V^n}), for the transformations of the components of a vector (\xi^{i'}=\dfrac{\partial x^{i'}}{\partial x^i}\xi^i) from ({A^n}) and of a covector (\xi_{i'}=\dfrac{\partial x^i}{\partial x^{i'}}\xi_i) from ({B^n}) entail the transformations

[
u^{i'}=
\frac{
\dfrac{\partial x^{i'}}{\partial x^i}u^i
}{
-\dfrac{1}{(n+1)}
\dfrac{\partial \ln \det |\partial x^{r'}/\partial x^r|}{\partial x^j}
u^j+1
},
]

[
u_{i'}=\frac{\partial x^i}{\partial x^{i'}}u_i
-\frac{1}{(n+1)}
\frac{\partial \ln \det |\partial x^{r'}/\partial x^r|}{\partial x^{i'}},
]

of the components of the corresponding objects from ({P^n}) and ({Q^n}).

The object ((u_0^i)) determines in each ({P^n}) an invariant point, and in each ({Q^n}) an invariant hyperplane

[
u_{0i}^i+1=0,
]

and the object ((u_i^0)) in ({P^n}) determines the invariant hyperplane

[
u_i^0 u^i + 1 = 0,
]

and in ({Q^n})—an invariant point.

To the sum of vectors ((\xi^i+\eta^i)) and covectors ((\xi_i+\eta_i)) there correspond, respectively, the objects

[
u^i \oplus v^i =
\frac{u^i+v^i+a_j(u^i v^j+u^j v^i)}
{-a_k a_l u^k v^l+1},
\tag{5}
]

[
u_i \oplus v_i = u_i+v_i-a_i,
\tag{6}
]

and to the product of a vector by a number ((\lambda\cdot u^i)) and of a covector by a number ((\lambda\cdot u_i)) the objects

[
\lambda \odot u^i =
\frac{\lambda u^i}{+a_j(1-\lambda)u^j+1},
\tag{7}
]

[
\lambda \odot u_i = \lambda u_i+a_i(1-\lambda).
\tag{8}
]

1. Let an affine connection be given on ({V^n}), ((\Gamma^p_{jk}=\Gamma^p_{kj})); then, substituting into the system

[
\frac{\partial \xi^i}{\partial x^k}+\xi^l \Gamma^i_{lk}=0,
]

which defines parallel displacement of the vector ((\xi^i)), the value of the latter from (3), we obtain

[
\frac{
\partial\left(\dfrac{u^i}{1+a_pu^p}\right)
}{
\partial x^k
}
+
\frac{u^l}{(1+a_pu^p)}\Gamma^i_{jk}=0
]

or

[
\left(\delta^i_q-\frac{a_q u^i}{(1+a_pu^p)}\right)
\frac{\partial u^q}{\partial x^k}
=
\left(
\frac{u^i}{1+a_pu^p}\frac{\partial a_j}{\partial x^k}
-\Gamma^i_{jk}
\right)u^j.
\tag{9}
]

The determinant of the obtained system is

[
\det\left|\delta^i_q-\frac{a_q u^i}{(1+a_pu^p)}\right|
=
\frac{1}{1+a_pu^p}\ne 0.
]

Solving (9) with respect to the derivatives (\partial u^q/\partial x^k), we shall have

[
\frac{\partial u^q}{\partial x^k}
=
-u^j\Gamma^q_{jk}
+
u^i u^q
\left(
\frac{\partial a_j}{\partial x^k}
-a_i\Gamma^i_{jk}
\right).
\tag{10}
]

This system defines projective displacements of the local centro-projective spaces ({P^n}) along a curve on the manifold ({V^n}).

Indeed, for every smooth curve

[
x^i=x^i(t)
\tag{11}
]

the system (10) gives a system of ordinary differential equations

[
\frac{du^q}{dt}
=
-u^j\Gamma^q_{jk}\frac{dx^k}{dt}
+
u^i u^q
\left(
\frac{\partial a_j}{\partial x^k}
-a_i\Gamma^i_{jk}
\right)
\frac{dx^k}{dt},
]

which, under the initial conditions (t=t_0,\ \left.u^q\right|_{u^q=u_0^q}), has a unique solution.

It should be noted that the projective displacement of ({P^n}) from the point (M_0) to the point (M_1) along the curve (11) can be carried out by successive application of the following three transformations: 1) transformation (3) at the point (M_0); 2) transformation of parallel translation in the given

of the affine connection from the point (M_0) to the point (M_1) along the curve (11); 3) transformation (1) at the point (M_1).

In particular, if the curve (11) is regarded as closed ((M_0 \equiv M_1)), then from the remark made above we obtain:

The holonomy group of the projective transports defined by the system (10) is a group similar to the corresponding holonomy group of the affine-connection space under consideration.

Next, considering the system (\dfrac{\partial \xi_i}{\partial x^k}-\xi_l\Gamma^l_{ik}=0) and substituting in place of ((\xi_i)) its value from (4), we obtain

[
\frac{\partial u_i}{\partial x^k}
=
u_l\Gamma^l_{ik}
+
\left(
\frac{\partial a_i}{\partial x^k}
-
a_l\Gamma^l_{ik}
\right).
]

Analogously to the preceding case, this system defines a transport of the spaces ({Q^n}) along the curve (11), and here again the holonomy group of these transports is similar to the holonomy group of the affine-connection space.

The operations (5) and (7), (6) and (8) are commutative with the operations of transport along a curve.

1. If the connection (\Gamma^p_{ik}) is Riemannian and (g_{ij}) is the fundamental metric tensor, then in each ({P^n}) there arises the invariant hyperquadric

[
[a_i a_j-g_{ij}]u^i u^j+2a_i u^i+1=0,
]

and the invariant hyperplane

[
a_i u^i+1=0,
]

defined by the object ((a_i)), becomes the polar hyperplane of the central point ((u^i=0)).

The system (10) now defines projective-metric transports. Moreover, all projective-metric transports of ({P^n}) that leave the point ((u^i=0)) invariant can always be defined by this system, since no restrictions are imposed on the components of the tensor ((g_{ij})) and of the object ((a_i)).

Let us also note that if one takes

[
a_i=-\frac{1}{(n+1)}\Gamma^a_{ai},
]

then the system (10) assumes the form

[
\frac{\partial u^q}{\partial x^k}
=
-u^j\Gamma^q_{jk}
-
\frac{1}{(n+1)}u^i u^q
\left(
\frac{\partial \Gamma^a_{aj}}{\partial x^k}
-
\Gamma^a_{ai}\Gamma^i_{jk}
\right),
\tag{12}
]

and the transport of the local centro-projective spaces ({P^n}) will be invariantly determined by the original affine-connection space. In addition, the object ((\Gamma^a_{ai})) will determine in each ({P^n}) the invariant hyperplane

[
-\frac{1}{(n+1)}\Gamma^a_{aj}u^j+1=0
]

and thus will acquire a concrete geometric meaning.

1. Suppose now that the Ricci tensor ((R_{ij})) of the affine-connection space under consideration has a nondegenerate symmetric part

[
\sigma_{ij}=\frac{R_{ij}+R_{ji}}{2};
\tag{13}
]

then in each ({P^n}) an invariant hyperquadric is determined

[
\left[
\frac{1}{(n+1)^2}\Gamma^a_{ai}\Gamma^b_{bj}
-
\frac{\sigma_{ij}}{n-1}
\right]u^i u^j
-
\frac{2}{(n+1)}\Gamma^a_{ai}u^i
+
1=0,
]

but, this time, the projective-metric transfers will be given not by system (12), but by the system

[
\frac{\partial u^q}{\partial x^k}
=
-u^i\widetilde{\Gamma}^{q}{ik}
-
\frac{1}{(n+1)}u^i u^q
\left(
\frac{\partial\Gamma^a}}{\partial x^k
-
\Gamma^a_{ai}\widetilde{\Gamma}^{i}_{jk}
\right),
\tag{14}
]

where (\widetilde{\Gamma}^{q}_{jk}) is the Riemannian connection constructed for the tensor (13).

It is not difficult to show that formulas (12) and (14) coincide if and only if the original space is a space of constant curvature.

1. In the latter case one can always pass to such a coordinate system in which

[
\Gamma^q_{jk}=\widetilde{\Gamma}^{q}{jk}
=
-
\frac{
\delta^q_j(cx^i+c_j)}x^i+c_k)+\delta^q_k(c_{ij
}{
c_{ij}x^i x^j+2c_i x^i+1
}
]

[
(c_{il},c_i,c=\mathrm{const})
\quad\text{and}\quad
\det|c_i c_j-c_{ij}|\ne 0.
]

This coordinate system is determined up to arbitrary fractional-linear transformations ((^2)), and equations (12) in these coordinates take the form

[
\frac{\partial u^q}{\partial x^k}
=
\frac{
u^i\left[\delta^q_j(c_{ik}x^i+c_k)+\delta^q_k(c_{ij}x^i+c_j)+c_{jk}u^q\right]
}{
c_{ij}x^i x^j+2c_i x^i+1
}.
\tag{15}
]

Similarly, for the object ((u_j)) we obtain

[
\frac{\partial u_j}{\partial x^k}
=
\frac{
-u_j(c_{kl}x^l+c_k)-u_k(c_{jl}x^l+c)+c_{jk}
}{
c_{il}x^i x^l+2c_i x^i+1
}.
\tag{16}
]

In conclusion we note that, if one does not assume (\det|c_i c_j-c_{ij}|\ne 0), then formulas (15) and (16) determine the transfer of the objects ((u^i)) and ((u_i)) for symmetric projective-Euclidean spaces ((^3)).

Moscow City Pedagogical Institute
named after V. P. Potemkin

Received
1 III 1960

CITED LITERATURE

(^1) V. G. Lemlein, DAN, 129, No. 2 (1959).
(^2) V. G. Lemlein, DAN, 131, No. 1 (1960).
(^3) P. A. Shirokov, Proceedings of the Seminar on Vector and Tensor Analysis, issue 8 (1950).