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

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

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

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

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

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).