MATHEMATICS

V. G. LEMLEIN

ON THE INDUCTION OF A CONNECTION OF CONSTANT CURVATURE IN ASSOCIATED CENTRO-PROJECTIVE SPACES OF A LOCALLY PROJECTIVE MANIFOLD

(Presented by Academician P. S. Aleksandrov on 16 XI 1959)

1. Let, in a locally projective manifold, i.e. in a manifold defined by nondegenerate fractional-linear transformations of local coordinate systems,

[
x^{i'}=\frac{a_i^{i'}x^i}{b_i x^i+1}+c^{i'}
\qquad
(a_i^{i'},\, b_i,\, c^{i'}=\mathrm{const}),
\tag{1}
]

there be given a principal relative scalar (a=a(x)) of weight one. Then the object

[
\gamma^p_{jk}(x)=
\frac{1}{(n+1)}\delta_j^p
\frac{\partial \ln a(x)}{\partial x^k}
+
\frac{1}{(n+1)}\delta_k^p
\frac{\partial \ln a(x)}{\partial x^j}
\tag{2}
]

may be taken as the object of an affine connection ((^{1})), while the curvature tensor and the Ricci tensor, as direct computations show, take, respectively, the form

[
R^h_{kij}(x)=
\frac{1}{(1-n)}
\left(\delta_i^h R_{jk}-\delta_j^h R_{ik}\right);
\tag{3}
]

[
R_{ij}(x)=
\frac{1-n}{n+1}
\left(
\frac{1}{(n+1)}
\frac{\partial \ln a}{\partial x^i}
\frac{\partial \ln a}{\partial x^j}
-
\frac{\partial^2 \ln a}{\partial x^i\partial x^j}
\right)
=
(1-n)\,a^{\frac{1}{n+1}}
\frac{\partial^2 a^{-\frac{1}{n+1}}}{\partial x^i\partial x^j}.
\tag{4}
]

In what follows we shall assume that

[
\det |R_{ij}(x)|\ne 0.
\tag{5}
]

1. The principal relative scalar in each local centro-projective space ({P^n}) ((^{2})) determines the quantity

[
\alpha(x,u)=
\frac{a(x)}
{\left(
\frac{a^{\frac{1}{n+1}}}{2}
\frac{\partial^2 a^{-\frac{2}{n+1}}}{\partial x^i\partial x^j}
u^i u^j
-
\frac{2}{(n+1)}
\frac{\partial \ln a}{\partial x^i}
u^i
+
1
\right)^{\frac{n+1}{2}}},
\tag{6}
]

which is transformed under fractional-linear 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^q}
u^q+1},
]

induced by the transformations (1) in ({P^n}), according to the law of a relative scalar of weight one and satisfying the conditions

[
\alpha(x,u)\big|{u=0}=a(x),\qquad
\frac{\partial \alpha(x,u)}{\partial u^k}\bigg|
=
\frac{\partial a(x)}{\partial x^k},\qquad
\frac{\partial \alpha(x,u)}{\partial u^k\partial u^l}\bigg|_{u=0}
=
\frac{\partial^2 a(x)}{\partial x^k\partial x^l}.
\tag{7}
]

From (\alpha) one can construct in ({P^n}) the connection

[
\Gamma^p_{jk}(x,u)=
]

[

\frac{
\delta^p_j\left(
\frac{1}{(n+1)}\frac{\partial\ln a}{\partial x^k}
-
\frac{a^{\frac{2}{n+1}}}{2}\frac{\partial^2 a^{-\frac{2}{n+1}}}{\partial x^k\partial x^q}u^q
\right)
+
\delta^p_k\left(
\frac{1}{(n+1)}\frac{\partial\ln a}{\partial x^j}
-
\frac{a^{\frac{2}{n+1}}}{2}\frac{\partial^2 a^{-\frac{2}{n+1}}}{\partial x^j\partial x^q}u^q
\right)
}{
\frac{a^{\frac{2}{n+1}}}{2}\frac{\partial^2 a^{-\frac{2}{n+1}}}{\partial x^q\partial x^l}u^q u^l
-
\frac{2}{(n+1)}\frac{\partial\ln a}{\partial x^q}u^q
+1
}.
\tag{8}
]

3. Theorem. In order that the connection defined in ({P^n}) by the object

[
\gamma^p_{jk}(u)=
\frac{1}{(n+1)}\delta^p_j\frac{\partial\ln\alpha}{\partial u^k}
+
\frac{1}{(n+1)}\delta^p_k\frac{\partial\ln\alpha}{\partial u^j},
\tag{9}
]

be a connection of constant curvature (K\ne 0), it is necessary and sufficient that it be Riemannian; and this is possible if and only if

[
\alpha=
\frac{c}{c_{ij}u^i u^j+2c_i u^i+1},
\qquad
\det|c_i c_j-c_{ij}|\ne 0.
\tag{10}
]

Moreover, under the natural requirement

[
\alpha^2=\det|g_{ij}|
\tag{11}
]

the curvature (K) and the metric tensor itself ((g_{ij}=g_{ji})) are determined up to factors (\varepsilon) (respectively (\bar\varepsilon)), equal to (n)-th roots of unity, and
[
c^2=\frac{1}{K^n}\det|c_i c_j-c_{ij}|.
]

Proof. The requirement of covariant constancy of the tensor (g_{ij}) with respect to the connection (9) leads to the system

[
\frac{\partial g_{ij}}{\partial u^k}
=
\frac{1}{(n+1)}
\left(
g_{ik}\frac{\partial\ln\alpha}{\partial u^j}
+
2g_{ij}\frac{\partial\ln\alpha}{\partial u^k}
+
g_{kj}\frac{\partial\ln\alpha}{\partial u^i}
\right),
\tag{12}
]

whose integrability conditions have the form

[
\begin{aligned}
&g_{ik}\left(
\frac{1}{(n+1)}\frac{\partial\ln\alpha}{\partial u^j}\frac{\partial\ln\alpha}{\partial u^l}
-
\frac{\partial^2\ln\alpha}{\partial u^j\partial u^l}
\right)
+
g_{kj}\left(
\frac{1}{(n+1)}\frac{\partial\ln\alpha}{\partial u^i}\frac{\partial\ln\alpha}{\partial u^l}
-
\frac{\partial^2\ln\alpha}{\partial u^i\partial u^l}
\right)
\
&\quad
-
g_{il}\left(
\frac{1}{(n+1)}\frac{\partial\ln\alpha}{\partial u^j}\frac{\partial\ln\alpha}{\partial u^k}
-
\frac{\partial^2\ln\alpha}{\partial u^j\partial u^k}
\right)
-
g_{lj}\left(
\frac{1}{(n+1)}\frac{\partial\ln\alpha}{\partial u^i}\frac{\partial\ln\alpha}{\partial u^k}
-
\frac{\partial^2\ln\alpha}{\partial u^i\partial u^k}
\right)
=0 .
\end{aligned}
\tag{13}
]

Hence, in particular, for (i=j) we have

[
g_{ik}
\left(
\frac{1}{(n+1)}\frac{\partial\ln\alpha}{\partial u^i}\frac{\partial\ln\alpha}{\partial u^l}
-
\frac{\partial^2\ln\alpha}{\partial u^i\partial u^l}
\right)
=
g_{il}
\left(
\frac{1}{(n+1)}\frac{\partial\ln\alpha}{\partial u^i}\frac{\partial\ln\alpha}{\partial u^k}
-
\frac{\partial^2\ln\alpha}{\partial u^i\partial u^k}
\right).
]

It is now not difficult to see that system (13) is equivalent to

[
\frac{1}{(n+1)}\,\frac{\partial \ln \alpha}{\partial u^i}\,
\frac{\partial \ln \alpha}{\partial u^j}
-\frac{\partial^2 \ln \alpha}{\partial u^i\partial u^j}
=-(n+1)K\cdot g_{ij}.
\tag{14}
]

Substituting into (12) the value of (g_{ij}) from (14), we obtain

[
-\frac{\partial \ln K}{\partial u^k}
\left(
\frac{1}{(n+1)}\,\frac{\partial \ln \alpha}{\partial u^i}
\frac{\partial \ln \alpha}{\partial u^j}
-\frac{\partial^2 \ln \alpha}{\partial u^i\partial u^j}
\right)
=
]

[

\frac{4}{(n+1)^2}\,
\frac{\partial \ln \alpha}{\partial u^i}
\frac{\partial \ln \alpha}{\partial u^j}
\frac{\partial \ln \alpha}{\partial u^k}
-\frac{2}{(n+1)}\,
\frac{\partial \ln \alpha}{\partial u^{(i}}
\frac{\partial^2 \ln \alpha}{\partial u^j\partial u^{k)}}
+\frac{\partial^3 \ln \alpha}{\partial u^i\partial u^j\partial u^k}.
\tag{15}
]

However, in fact equations (15) can be written in the simpler form

[
\frac{\partial^3 \ln \alpha}{\partial u^i\partial u^j\partial u^k}
-\frac{2}{(n+1)}\,
\frac{\partial \ln \alpha}{\partial u^{(i}}
\frac{\partial^2 \ln \alpha}{\partial u^i\partial u^j)}
+\frac{4}{(n+1)^2}\,
\frac{\partial \ln \alpha}{\partial u^i}
\frac{\partial \ln \alpha}{\partial u^j}
\frac{\partial \ln \alpha}{\partial u^k},
\tag{16}
]

for (K=\mathrm{const}). Indeed, by formulas (3) and (4), written for the connection (9), we have

[
R_{ij}(u)=(n-1)K\cdot g_{ij},\qquad
R_{lk,ij}(u)=K\cdot (g_{lj}g_{ki}-g_{li}g_{kj}).
\tag{17}
]

Thus, for (n>2), (K=\mathrm{const}).

Further, for (n=2), by virtue of the symmetry of the right-hand side of (15) with respect to any pair of indices, the equalities

[
\left(
\frac{1}{(n+1)}\frac{\partial \ln \alpha}{\partial u^1}
\frac{\partial \ln \alpha}{\partial u^1}
-\frac{\partial^2 \ln \alpha}{(\partial u^1)^2}
\right)
\frac{\partial \ln K}{\partial u^2}
-
\left(
\frac{1}{(n+1)}\frac{\partial \ln \alpha}{\partial u^1}
\frac{\partial \ln \alpha}{\partial u^2}
-\frac{\partial^2 \ln \alpha}{\partial u^1\partial u^2}
\right)
\frac{\partial \ln K}{\partial u^1}
=0,
]

[
-\left(
\frac{1}{(n+1)}\frac{\partial \ln \alpha}{\partial u^2}
\frac{\partial \ln \alpha}{\partial u^1}
-\frac{\partial^2 \ln \alpha}{\partial u^2\partial u^1}
\right)
\frac{\partial \ln K}{\partial u^2}
-
\left(
\frac{1}{(n+1)}\frac{\partial \ln \alpha}{\partial u^2}
\frac{\partial \ln \alpha}{\partial u^2}
-\frac{\partial^2 \ln \alpha}{(\partial u^2)^2}
\right)
\frac{\partial \ln K}{\partial u^1}
=0
]

must hold. Consequently, (K=\mathrm{const}), since (\det|R_{ij}(u)|\ne 0).

Solving now system (16), which can be rewritten in the form

[
\frac{
\partial\left(
\frac{\partial \ln \alpha}{\partial u^i}\,\alpha^{-\frac{2}{n+1}}
\frac{\partial \ln \alpha}{\partial u^j}\,\alpha^{-\frac{2}{n+1}}
\right)
}{\partial u^k}
+
\frac{
\partial\left(
\frac{\partial^2 \ln \alpha}{\partial u^i\partial u^j}\,
\alpha^{-\frac{2}{n+1}}
\right)
}{\partial u^k}
=0
]

or

[
\frac{\partial^2 \alpha^{-\frac{2}{(n+1)}}}{\partial u^i\partial u^j}
=\widetilde c_{ij},
]

where (\widetilde c_{ij}=\mathrm{const}) and (\widetilde c_{ij}=\widetilde c_{ji}), we obtain

[
\alpha=
\frac{c}{
\left(c_{ij}u^i u^j+2c_i u^i+1\right)^{\frac{n+1}{2}}
}.
\tag{18}
]

Further, from (17) we have

[
\det|R_{ij}(u)|=(n-1)^n\cdot K^n\cdot \det|g_{ij}|,
]

therefore, by virtue of conditions (11) and (10), we obtain

[
c^2=\frac{1}{K^n}\cdot \det|c_i c_j-c_{ij}|.
]

It is also clear from this that the curvature (K) and the metric tensor (g_{ij}) are determined up to factors (\varepsilon) (respectively (\varepsilon)).

1. The connection (8) is obtained by substituting (6) into (9), and since (6) has the same structure as (18), while, by virtue of (5),

[
\det |R_{ij}(x,u)|=(n-1)^n K^n(x)\det |g_{ij}(x,u)|\ne 0,
\tag{19}
]

then (8) defines in each ({P^n}) a connection of constant curvature. Further, by virtue of (7),

[
\begin{aligned}
\Gamma^p_{jk}(x,u)\big|{u=0}&=\gamma^p(x),&
\frac{\partial \Gamma^p_{jk}(x,u)}{\partial u^l}\bigg|{u=0}
&=\frac{\partial \gamma^p,\}(x)}{\partial x^l
R^h{}{kij}(x,u)\big|}&=R^h{{kij}(x),&
R(x,u)\big|{u=0}&=R(x).
\end{aligned}
\tag{20}
]

Finally, under the natural requirement

[
\alpha^2(x,u)=\det |g_{ij}(x,u)|
]

from (19), taking (20) into account, we have

[
K(x)=\frac{\varepsilon}{(n-1)}\sqrt[n]{\frac{\det |R_{ij}(x)|}{a^2(x)}},
\tag{21}
]

and, consequently, from

[
R_{ij}(x,u)=(n-1)K(x)g_{ij}(x,u)
]

we obtain

[
g_{ij}(x,u)=\bar{\varepsilon}R_{ij}(x,u)\sqrt[n]{\frac{a^2(x)}{\det |R_{ij}(x)|}},
]

and for (u=0)

[
g_{ij}=(x)\bar{\varepsilon}R_{ij}(x)\sqrt[n]{\frac{a^2(x)}{\det |R_{ij}(x)|}}.
\tag{22}
]

Let us note in conclusion that the curvature (21) is a function of the point of the locally projective manifold and, consequently, when (K\ne \mathrm{const}), the covariant derivative of the tensor (22) with respect to the connection (2) is different from zero.

Moscow City Pedagogical Institute
named after V. P. Potemkin

Received
13 XI 1959

References Cited

(^{1}) V. G. Lemlein, DAN, 128, No. 4 (1959).
(^{2}) V. G. Lemlein, DAN, 129, No. 2 (1959).