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For a general $n+1$ manifold $\tilde{M}$ of constant curvature $k$, the curvature tensor has the form
$\tilde{R}(\tilde{X}, \tilde{Y})\tilde{Z} = k ( \tilde{g}(\tilde{X},\tilde{Z})\tilde{Y} - \tilde{g}(\tilde{X},\tilde{Z})\tilde{Y})$
With a hypersurface $M$ the second fundamental form $h$ shows itself in the change in covariant differentiations:
$\tilde{\nabla}_XY = \nabla_XY + h(X,Y)$