The authors investigate the curvature properties of indefinite almost contact manifolds \((M,\varphi,\xi,\) \(\eta,g).\) Formally, \((\varphi,\xi,\eta,g)\) is an almost contact metric structure on the differentiable manifold \(M\) [cf., e.g., \textit{D. E. Blair} [Contact manifolds in Riemannian geometry. Lect. Notes Math. 509 (1976; Zbl 0319.53026)] but the metric \(g\) is not assumed to be positive definite. The investigation focuses on the so-called \(C(\alpha)\)-manifolds introduced by \textit{D. Janssens} and \textit{L. Vanhecke} [Kodai Math. J. 4, 1-27 (1981; Zbl 0472.53043)], i.e., those satisfying the condition \[ R(X,Y,Z,W-R(X,Y,\varphi Z,\varphi W)= \] \[ \alpha\bigl(g(Y,Z)g(X,W)-g(X,Z)g(Y,W)-g(Y,\varphi Z)g(X,\varphi W) +g(X,\varphi Z)g(Y,\varphi W)\bigr) \] for any vector fields, \(\alpha\) being a constant. By means of the study of the Jacobi operator along spacelike, timelike and null geodesics, spaces of constant curvature are characterized as well as spaces of pointwise constant \(\varphi\)-sectional curvature. There is an essential difference on the behaviour of the Jacobi operator along null and non-null geodesics. This motivates the definition of the \(\varphi\)-isotropic \(C(\alpha)\)-manifolds as those satisfying the condition \(R(U,\varphi U)\varphi U=c_UU\) (\(c_U=\text{const.}\)) for any null vector \(U\) tangent to the contact distribution. A local classification of \(\varphi\)-isotropic \(C(\alpha)\)-manifolds with parallel and diagonalizable Ricci tensor is obtained.