Abstract Generated abstract
This paper studies Riemannian spaces admitting a generally recurrent symmetric tensor of second order, a condition encompassing recurrent tensors, tensors associated with hyperquadrics, common geodesic metrics, and projective vector fields. By analyzing the compatibility of the defining differential system through the eigenvalue decomposition of the tensor, the author derives normal forms for the metric and the tensor in coordinates adapted to the distinct proper values. The main result states that, for \(n \geq 4\) and at least four distinct proper values, such a space admits the tensor if and only if its metric is conformal to a Levi-Civita metric. Several special cases recover known results on recurrent tensors, hypersurfaces of second order, and projective transformations, and extend earlier formulas to multiple curvatures.
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
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