Full Text
UDC 513
MATHEMATICS
V. S. SOBCHUK
ON RIEMANNIAN SPACES ADMITTING A GENERALLY RECURRENT SYMMETRIC TENSOR OF THE SECOND ORDER
(Presented by Academician A. D. Aleksandrov on 11 IX 1968)
1°. Definition. A symmetric tensor \(b_{ij}\) of the second order will be called generally recurrent if it satisfies the condition
\[ b_{ij,l}=\bar{\lambda}_{l}g_{ij}+\lambda_i g_{jl}+\lambda_j g_{il}+\bar{\mu}_{l}b_{ij}+\mu_i b_{jl}+\mu_j b_{il}, \tag{1} \]
where \(g_{ij}\) is the metric tensor of the Riemannian space \(V_n\); \(\bar{\lambda}_i,\lambda_i,\bar{\mu}_i,\mu_i\) are certain covariant tensors, which we shall call the tensors of generalized recurrence; the comma denotes covariant differentiation.
In particular, if \(\bar{\lambda}_i=0,\lambda_i=0,\mu_i=0\), then we obtain a recurrent tensor \(b_{ij}\) \((^1)\). Condition (1) also includes the tensor characteristic of hypersurfaces of the second order \((^2)\) (for \(\bar{\lambda}_i=0,\lambda_i=0,\bar{\mu}_i=\mu_i\)), and the condition that the metrics \(g_{ij}\) and \(b_{ij}\) have common geodesics \((^3)\) (for \(\lambda_i=0,\bar{\lambda}_i=0,\bar{\mu}_i=2\mu_i\)), and, finally, the characteristic condition that the vector field \(\xi_i\) determines a one-parameter group of projective transformations \((^3)\) (for \(\bar{\mu}_i=0,\mu_i=0,\bar{\lambda}_i=2\lambda_i,\ b_{ij}=\xi_{(i,j)}\)).
In the present article we find all Riemannian spaces admitting a generally recurrent symmetric tensor of the second order, i.e., we find all metrics for which the system of equations (1) is compatible.
2°. Preliminary information. Among the eigenvalues \(k_1,k_2,\ldots,k_n\) of the tensor \(b_{ij}\), i.e., among the roots of the equation \(\lvert kg_{ij}-b_{ij}\rvert=0\), there may also be multiple ones. Let a frame of eigenvectors \(\eta_{a_1}^{\,i}\) be orthonormalized: \(g_{ij}\eta_{a_1}^{\,i}\eta_{b_1}^{\,j}=\delta_{ab}\); then we also have \(b_{ij}\eta_{a_1}^{\,i}\eta_{b_1}^{\,j}=k_a\delta_{ab}\), so that from (1) we obtain
\[ b_{ij,l}\eta_{a_1}^{\,i}\eta_{b_1}^{\,j}\eta_{c_1}^{\,l}=0,\qquad a,b,c\ne . \tag{2} \]
When condition (2) is fulfilled, as is known \((^3)\), one can pass to new variables in such a way that we shall have
\[ g_{ij}du^i du^j=\sum_{\alpha=1}^{p}\Phi_\alpha,\qquad b_{ij}du^i du^j=\sum_{\alpha=1}^{p}k_\alpha\Phi_\alpha, \tag{3} \]
where \(k_1,\ldots,k_p\) are distinct roots, and the form \(\Phi_\alpha\) contains only the differentials of the variables \(u^{i_\alpha}\) corresponding to \(k_\alpha\). The variables \(u^1,u^2,\ldots,u^n\) are divided into \(p\) groups \(u^{i_1},\ldots,u^{i_p}\) (according to the number of distinct roots), and the number of variables in each group is equal to the multiplicity of the corresponding root, for example \(u^{i_1}\equiv(u^1,\ldots,u^{m_1})\), where \(m_1\) is the multiplicity of the root \(k_1\). If among the roots \(k_1,\ldots,k_n\) there are simple ones, then we write them in the first places. Thus, if \(k_\alpha\) is a simple root, then the group of variables \(u^{i_\alpha}\) consists of the single variable \(u^\alpha\).
Equalities (3) mean that
\[ g_{i_\alpha j_\beta}=b_{i_\alpha j_\beta}=0,\ \alpha\ne\beta;\qquad b_{i_\alpha j_\alpha}=k_\alpha g_{i_\alpha j_\alpha}, \tag{4} \]
where \(g_{i_\alpha j_\alpha}\), \(b_{i_\alpha j_\alpha}\) depend, generally speaking, on all variables \(u^1,\ldots,u^n\).
3°. Basic equations. Computing directly the derivative \(b_{i_\alpha j_\alpha,l_\beta}\), and also from (1) (using in both cases (4)), we find
\[ (k_\alpha-k_\beta)\partial\ln|g_{j_\beta,l_\beta}|/\partial u^{i_\alpha} =2\lambda_{i_\alpha}+2k_\beta\mu_{i_\alpha}. \tag{5} \]
Similarly, computing \(b_{j_\beta l_\beta,i_\alpha}\) and \(b_{i_\alpha i_\alpha,i_\alpha}\), we obtain
\[ \partial k_\beta/\partial u^{i_\alpha}=\bar\lambda_{i_\alpha}+k_\beta\bar\mu_{i_\alpha}, \tag{6} \]
\[ \partial k_\alpha/\partial u^{i_\alpha} =\bar\lambda_{i_\alpha}+k_\alpha\bar\mu_{i_\alpha} +2\lambda_{i_\alpha}+2k_\alpha\mu_{i_\alpha}. \]
Computing \(b_{i_\alpha i_\alpha,j_\alpha}\), using (1), (4), and (7), we have
\[ g_{i_\alpha i_\alpha}(\lambda_{j_\alpha}+k_\alpha\mu_{j_\alpha}) -g_{i_\alpha j_\alpha}(\lambda_{i_\alpha}+k_\alpha\mu_{i_\alpha})=0. \tag{7} \]
Interchanging the indices \(i_\alpha\) and \(j_\alpha\) and using the positive definiteness of the form \(g_{ij}du^i du^j\), we find
\[ \lambda_{i_\alpha}+k_\alpha\mu_{i_\alpha}=0,\quad \text{if } k_\alpha \text{ is multiple.} \tag{8} \]
Consequently,
\[ \partial k_\alpha/\partial u^{i_\alpha} =\bar\lambda_{i_\alpha}+k_\alpha\bar\mu_{i_\alpha}, \quad \text{if } k_\alpha \text{ is multiple.} \tag{9} \]
From (6), replacing \(\beta\) by \(\gamma\) and subtracting from (6), we obtain
\[ \partial\ln|k_\beta-k_\gamma|/\partial u^{i_\alpha} =\bar\mu_{i_\alpha}. \tag{10} \]
From (10), replacing \(\gamma\) by \(\delta\) and subtracting from (10), we obtain
\[ \frac{\partial}{\partial u^{i_\alpha}} \ln\left|(k_\beta-k_\gamma)/(k_\beta-k_\delta)\right|=0, \quad \alpha,\beta,\gamma,\delta \ne . \tag{11} \]
Here and below \(p\ge 4\). Taking the logarithm and differentiating with respect to \(u^{l_\beta}\) of the identity
\[ \frac{k_\alpha-k_\beta}{k_\alpha-k_\gamma} = \frac{k_\alpha-k_\beta}{k_\alpha-k_\delta} \frac{k_\alpha-k_\delta}{k_\alpha-k_\gamma} \]
and taking (11) into account, we find
\[ \frac{\partial}{\partial u^{l_\beta}} \ln\left|\frac{k_\alpha-k_\beta}{k_\alpha-k_\gamma}\right| = \frac{\partial}{\partial u^{l_\beta}} \ln\left|\frac{k_\alpha-k_\beta}{k_\alpha-k_\delta}\right| \equiv \varphi_{\alpha l_\beta}(u^{i_\alpha},u^{l_\beta}). \tag{12} \]
4°. The functions \(\varphi_{\beta i_\alpha}\). Eliminating \(\bar\lambda_{i_\alpha}\) from (6) and (9), we obtain
\[ \partial\ln|k_\beta-k_\alpha|/\partial u^{i_\alpha} =\bar\mu_{i_\alpha},\quad k_\alpha \text{ multiple.} \tag{13} \]
From (10) and (13) we find
\[ \frac{\partial}{\partial u^{i_\alpha}} \ln\left|\frac{k_\beta-k_\alpha}{k_\beta-k_\gamma}\right|=0, \quad \text{if } k_\alpha \text{ is multiple,} \]
i.e., we have \(\varphi_{\beta i_\alpha}=0\), if \(k_\alpha\) is multiple. Let \(\varphi_{\beta\alpha}\ne0\). Computing the ratio \(\varphi_{\gamma\alpha}:\varphi_{\beta\alpha}\) by means of (12) and replacing the derivatives of \(k_\alpha,k_\beta,k_\gamma\) by formulas (6) and (7), we obtain
\[ \varphi_{\gamma\alpha}/\varphi_{\beta\alpha} =(k_\alpha-k_\beta)/(k_\alpha-k_\gamma). \tag{14} \]
Substituting (14) into (12), we find
\[ \frac{\partial\varphi_{\beta\alpha}}{\partial u^{l_\beta}} =-\varphi_{\beta\alpha}\varphi_{\alpha l_\beta}. \tag{15} \]
The solution of the system (15) is the functions
\[ \varphi_{\beta\alpha}=\varphi_\alpha'/(\varphi_\alpha-\varphi_\beta), \tag{16} \]
where \(\varphi_\alpha\) is an arbitrary function of \(u_\alpha\), and \(\varphi_\alpha=\mathrm{const}\) if \(k_\alpha\) is multiple. Thus the functions \(\varphi_{\beta i_\alpha}\) are found from (16) both in the case when \(k_\alpha,k_\beta\) are multiple, and when they are simple.
5°. The curvatures \(k_\alpha\). From (14) and (16) we obtain
\[ (k_\alpha-k_\beta)/(\varphi_\alpha-\varphi_\beta) = (k_\alpha-k_\gamma)/(\varphi_\alpha-\varphi_\gamma), \tag{17} \]
i.e. the ratio \((k_\alpha-k_\beta)/(\varphi_\alpha-\varphi_\beta)\) does not depend on \(\beta\). But
\[
(k_\alpha-k_\beta)/(\varphi_\alpha-\varphi_\beta)
=(k_\beta-k_\alpha)/(\varphi_\beta-\varphi_\alpha),
\]
i.e., this ratio does not depend on \(\alpha\), and we shall denote it by \(F\):
\[ (k_\alpha-k_\beta)/(\varphi_\alpha-\varphi_\beta)=F. \tag{18} \]
From (18) we find \(k_\alpha-\varphi_\alpha F=k_\beta-\varphi_\beta F\equiv \Phi\), or
\[ k_\alpha=\varphi_\alpha F+\Phi, \tag{19} \]
where \(F\) and \(\Phi\) are arbitrary functions.
\(6^\circ\). Metric. Adding (5), (6) and subtracting (7), we obtain
\[ \frac{\partial}{\partial u^{i_\alpha}} \ln \left|\frac{g_{j_\beta l_\beta}}{g_{j_\gamma l_\gamma}}\right| =-\bar\mu_{i_\alpha}-2\mu_{i_\alpha}. \tag{20} \]
From (20), replacing \(\beta\) by \(\gamma\) and subtracting from (20), we obtain
\[ \frac{\partial}{\partial u^{i_\alpha}} \ln \left|\frac{g_{j_\beta l_\beta}}{g_{j_\gamma l_\gamma}}\right| = \frac{\partial}{\partial u^{i_\alpha}} \ln \left|\frac{k_\alpha-k_\beta}{k_\alpha-k_\gamma}\right|. \tag{21} \]
From (17) and (21) we find
\[ \frac{\partial}{\partial u^{i_\alpha}} \ln \left|\frac{g_{j_\beta l_\beta}}{g_{j_\gamma l_\gamma}}\right| = \frac{\partial}{\partial u^{i_\alpha}} \ln \left|\frac{\varphi_\alpha-\varphi_\beta}{\varphi_\alpha-\varphi_\gamma}\right|, \qquad \alpha\ne\beta,\gamma. \tag{22} \]
Integrating (22), we have
\[ \frac{g_{j_\beta l_\beta}}{g_{j_\gamma l_\gamma}} = B_{j_\beta l_\beta j_\gamma l_\gamma} \prod_{\sigma\ne\beta,\gamma} \left|\frac{\varphi_\sigma-\varphi_\beta}{\varphi_\sigma-\varphi_\gamma}\right|, \tag{23} \]
where \(B_{j_\beta l_\beta j_\gamma l_\gamma}\) are functions only of \(u^{k_\beta}, y^{m_\gamma}\). From (23) we obtain
\[ B_{j_\beta l_\beta j_\gamma l_\gamma} = B_{j_\beta l_\beta j_\delta l_\delta} B_{j_\delta l_\delta j_\gamma l_\gamma}. \]
Consequently,
\[ B_{j_\beta l_\beta j_\gamma l_\gamma} = B_{j_\beta l_\beta}/B_{j_\gamma l_\gamma}, \tag{24} \]
where \(B_{j_\beta l_\beta}\) are functions only of \(u^{m_\beta}\). From (23) and (24) we find
\[ g_{j_\beta l_\beta}/B_{j_\beta l_\beta} \prod_{\sigma\ne\beta}|\varphi_\sigma-\varphi_\beta| = g_{j_\gamma l_\gamma}/B_{j_\gamma l_\gamma} \prod_{\sigma\ne\gamma}|\varphi_\sigma-\varphi_\gamma| =A. \]
Thus, we obtain
\[ g_{j_\beta l_\beta} = A B_{j_\beta l_\beta} \prod_{\sigma\ne\beta}|\varphi_\sigma-\varphi_\beta|. \tag{25} \]
From (3) and (25) we find
\[ g_{ij}du^i du^j = A\sum_{\alpha=1}^{p} \prod_{\sigma\ne\alpha}|\varphi_\sigma-\varphi_\alpha|\, ds_\alpha^2, \tag{26} \]
\[ b_{ij}du^i du^j = A\sum_{\alpha=1}^{p} (\varphi_\alpha F+\Phi) \prod_{\sigma\ne\alpha}|\varphi_\sigma-\varphi_\alpha|\, ds_\alpha^2, \tag{27} \]
where \(A>0\), \(F\), \(\Phi\) are arbitrary functions; \(\varphi_\alpha\) are functions only of \(u^\alpha\), and \(\varphi_\alpha=\mathrm{const}\) if \(k_\alpha\) is multiple, \(\varphi_\alpha\ne\varphi_\beta\), \(\alpha\ne\beta\).
The metric
\[ ds^2=\sum_{\alpha=1}^{p} \prod_{\sigma\ne\alpha}|\varphi_\sigma-\varphi_\alpha|\, ds_\alpha^2 \]
is called the Levi-Civita metric \((^3)\). Consequently, the metric (26) is conformal to the Levi-Civita metric. If, in a space with metric (26), the tensor \(b_{ij}\) is chosen according to (27), then it will evidently be generalized recurrent. Thus the following theorem has been proved.
Theorem. A Riemannian space \(V_n\), \(n\geqslant 4\), admits a generalized recurrent symmetric tensor of the second order having \(p\) \((p\geqslant 4)\) distinct proper values if and only if \(V_n\) is conformal to a Levi-Civita space.
7. Some special cases.
From (10) and (19) we find \(\bar{\mu}_i=\partial\ln|F|/\partial u^i\); hence, and from (6), we obtain \(\bar{\lambda}_i=\partial\Phi/\partial u^i-\Phi\,\partial\ln|F|/\partial u^i\). From (5) and (7), taking into account the expressions found for \(\bar{\lambda}_i\) and \(\bar{\mu}_i\), we have
\[ 2\mu_i=-\partial\ln A/\partial u^i,\qquad 2\lambda_{i_\alpha}=\varphi'_\alpha F+(F\varphi_\alpha+\Phi)\partial\ln A/\partial u^{i_\alpha}. \]
1) A semirecurrent tensor, i.e. \(\lambda_i=0,\ \mu_i=0\). We find \(A=\mathrm{const}\), \(\varphi_\alpha=\mathrm{const}\). Consequently, the metric \(V_n\) is reducible. In particular, for a recurrent tensor we also have \(\bar{\lambda}_i=0\), since \(\Phi=cF,\ c=\mathrm{const}\). Consequently, \(k_\alpha=F(\varphi_\alpha+c)\)—Datta’s theorem \((^1)\) on the proportionality to one another of the proper values of a recurrent tensor.
2) If \(g_{ij}, b_{ij}\) are the first and second fundamental tensors of a hypersurface, then for \(\lambda_i=0,\ \bar{\lambda}_i=0,\ \bar{\mu}_i=\mu_i\) we obtain the tensor characteristic of a hypersurface \((^2)\). We find \(\Phi=cF,\ c=\mathrm{const},\ A=\prod_{\sigma=1}^{p} f_\sigma^{-1},\ f_\alpha=\varphi_\sigma+c,\)
\[ F=A^{-1/2}=\prod_{\sigma=1}^{p} f_\sigma^{1/2}. \]
Consequently,
\[ k_\alpha=f_\alpha\prod_{\sigma=1}^{p} f_\sigma^{1/2},\qquad g_{i_\alpha j_\alpha}=B_{i_\alpha j_\alpha}(u^{m_\alpha})\prod_{\sigma\ne\alpha}\left|1-\frac{f_\alpha}{f_\sigma}\right|, \]
i.e. we obtain the principal curvatures and the metric of a hyperquadric. At the same time we find
\[ \mu_{i_\alpha}=\frac{1}{p+2}\,\frac{\partial\ln|K|}{\partial u^{i_\alpha}},\qquad K=\prod_{\sigma=1}^{p} k_\sigma. \]
3) For \(\mu_i=0,\ \bar{\mu}_i=0,\ \bar{\lambda}_i=2\lambda_i\), we obtain \(F=\mathrm{const},\ A=\mathrm{const},\)
\[ \Phi=F\sum_{\sigma=1}^{p}\varphi_\sigma. \]
Consequently,
\[ k_\alpha=f_\alpha+\sum_{\sigma=1}^{p} f_\sigma,\qquad f_\alpha=F\varphi_\alpha,\qquad ds^2=\sum_{\alpha=1}^{p}\prod_{\sigma\ne\alpha}|f_\sigma-f_\alpha|\,ds_\alpha^2. \]
\[ b_{ij}du^i du^j =\sum_{\alpha=1}^{p}\left(f_\alpha+\sum_{\sigma=1}^{p} f_\sigma\right) \prod_{\sigma\ne\alpha}|f_\sigma-f_\alpha|\,ds_\alpha^2,\qquad \lambda_{i_\alpha} =\frac{\partial}{\partial u^{i_\alpha}}\left(\frac12\sum_{\sigma=1}^{p} f_\sigma\right). \]
Thus, we obtain the Levi-Civita metric.
4) For \(\bar{\lambda}_i=\lambda_i,\ \bar{\mu}_i=\mu_i\) we obtain \(A=F^{-2},\ \partial\Phi/\partial u^{i_\alpha}=\frac12\varphi'_\alpha F-\varphi_\alpha \partial F/\partial u^{i_\alpha}\), i.e. a generalization of our results \((^4)\) to the case of multiple curvatures.
Chernivtsi State University
Received
2 IX 1968
References
- D. K. Datta, Tensor, 15, No. 1, 61 (1964).
- L. L. Verbickii, Tr. seminara po vektorn. i tenzorn. analizu, vol. 7, 1949.
- A. S. Solodovnikov, UMN, 11, no. 4 (70), 45 (1956).
- V. S. Sobchuk, Tr. seminara po vektorn. i tenzorn. analizu, vol. 15, 1968.