Let M be an n-dimensional semi-Riemannian manifold with metric g of arbitrary index and the Levi-Civita connection D determined by g. A linear connection \(\tilde D\) on M is said to be a semi-symmetric metric connection if there exists a 1-form p such that the torsion tensor T of \(\tilde D\) is given by \(T(X,Y)=p(Y)X-p(X)Y\) and \(\tilde Dg=0.\)
The following theorems are proved here: If M admits \(\tilde D\) such that the Ricci tensors Ric of D and \(\tilde Ric\) of D are related by \(\tilde Ric=\phi\;Ric\), \(\phi\) being a function, then for \(n>3\), the curvature tensors K of D and \(\tilde K\) of \(\tilde D\) have the property \(\tilde K=\phi K+(1+\phi)C\), where C denotes the Weyl conformal curvature of D. Let \(\xi\) be an affine conformal vector field on M. Then \(\xi\) reduces to a homothetic vector field and \(LP=-AP\), if and only if \(\tilde D\) is invariant under the 1-parameter group of transformations generated by P, where L is the Lie derivative with respect to \(\xi\), \(A=const.\), and P is the vector field defined by \(g(P,X)=p(X)\). A geodesic U with respect to D is preserved by \(\tilde D\) if and only if U is null and orthogonal to the integral curves of P or U is an integral curve of P. (There has been a trend of showing an interplay between positive definite and indefinite Riemannian geometry, i.e. B. O’Neill [Semi-Riemannian geometry. With applications to relativity (1983; Zbl 0531.53051)], and the motivation of this paper may be to present some new contribution to this trend.)