# zbMATH — the first resource for mathematics

On quasi-conformally flat almost pseudo Ricci symmetric manifolds. (English) Zbl 1225.53016
This paper, as printed, is awfully distorted by many misprints, and the reviewer is surprised how this could come about during the referee procedure. However, after some frustrating efforts, an internet version cropped up [http://www1.au.edu.tw/ox_view/edu/tojms/j_paper/ Full_text/Vol-26/No-2/26(2)8-6(203-219).pdf ] which seems more accurate and which will be reviewed here instead.
The authors study, on Riemannian manifolds of dimension $$n > 3$$, several conditions on the Ricci tensor and on the conformal curvature tensors. The condition on the Ricci tensor $$S$$, named ‘almost pseudo Ricci-symmetric’, is
$(\nabla_{X} S)(Y,Z) = (A(X)+B(X))S(Y,Z)+A(Y)S(X,Z)+A(Z)S(Y,X),$
where $$A$$ and $$B$$ are nowhere vanishing Pfaffian forms. The condition on the conformal curvature tensors, named ‘quasi-conformally flat’, is $$W = 0$$ with
$W (X, Y )Z := -(n - 2)bC(X,Y)Z +(a + (n - 2)b) \tilde{C}(X,Y)Z,$
where $$a, b$$ are arbitrary constants not simultaneously zero and $$C,\tilde{C}$$ are, respectively, the conformal and the concircular curvature tensor. See K. Amur and Y. B. Maralabhavi [Tensor, New Ser. 31, 194–198 (1977; Zbl 0362.53010)].
Typical results (out of about $$11$$) are, e.g., Theorem 3.4: Every quasi-conformally flat almost pseudo Ricci-symmetric manifold with $$a \neq 0$$, $$a + b \neq 0$$ and $$a+(n-2)b \neq 0$$ is a manifold of quasi-constant curvature in the sense of B.-Y. Chen and K. Yano [Tensor, New Ser. 26, 318–322 (1972; Zbl 0257.53027)]. Theorem 3.7: A non-Einstein quasi-conformally almost pseudo Ricci-symmetric manifold of non-constant scalar curvature with $$2a -(n-1)(n-4)b \neq 0$$, $$a+(n-2)b \neq 0$$ and $$a+b \neq 0$$ is a subprojective manifold in the sense of Kagan (see [T. Adati, Tôhoku Math. J., II. Ser. 3, 330–342, 343–358 (1951); Tensor, n. Ser. 1, 105–115, 116–129, 130–136 (1951; Zbl 0045.11202)]).

##### MSC:
 53B20 Local Riemannian geometry