= Solution
\b[Curvature contractions.] Use the index contraction stipulated in the question. In three dimensions,
$$
R_{jk}=g^{il}K(g_{ik}g_{jl}-g_{il}g_{jk})=K(g_{jk}-3g_{jk})=-2Kg_{jk}.
$$
The same contraction in four dimensions gives
$$
R_{\beta\gamma}=-\frac\Lambda3(g_{\beta\gamma}-4g_{\beta\gamma})=\Lambda g_{\beta\gamma}.
$$
Then the <Ricci scalar> is $R=4\Lambda$ and the <Einstein tensor> is $G_{\mu\nu}=-\Lambda g_{\mu\nu}$. Thus $R_{\mu\nu}=\Lambda g_{\mu\nu}$ is precisely the vacuum <Einstein field equations> $G_{\mu\nu}+\Lambda g_{\mu\nu}=0$.
\b[The static solution.] For $A=1+\lambda^2r^2$, $A'=2\lambda^2r$ and $A''=2\lambda^2$. The given <Ricci tensor> components become
$$
R_{00}=-3\lambda^2A,\quad R_{11}=\frac{3\lambda^2}{A},\quad
R_{22}=3\lambda^2r^2,\quad R_{33}=3\lambda^2r^2\sin^2\theta.
$$
Each equals $-3\lambda^2$ times the corresponding <metric tensor> component. Therefore
$$
\boxed{\Lambda=-3\lambda^2.}
$$
This is <Anti-de Sitter spacetime> in <static coordinates on anti-de Sitter spacetime>, with curvature radius $1/\lambda$. The overall <metric signature> is opposite to the convention used in some descriptions of those coordinates; the geodesic constants below use the question's $(+,-,-,-)$ convention.
\b[Conserved quantities and radial motion.] In the equatorial plane, the <geodesic Lagrangian> is $L=A\dot t^2-\dot r^2/A-r^2\dot\phi^2$. Its <cyclic coordinates> give
$$
\boxed{k=A\dot t,\qquad h=r^2\dot\phi.}
$$
For rest <mass> $m$, $mk=p_t$ is the conserved Killing <energy> and $mh=-p_\phi$ is the axial <angular momentum>. Timelike normalization $L=1$ gives
$$
\boxed{\dot r^2=k^2-A\left(1+\frac{h^2}{r^2}\right)
=k^2-1-\lambda^2h^2-\lambda^2r^2-\frac{h^2}{r^2}.}
$$
For purely radial motion $h=0$, differentiating away from a turning point gives $\ddot r=-\lambda^2r$, and continuity extends that equation through the turning point. An outward launch from the origin at $\tau=0$ has
$$
\boxed{r(\tau)=\frac{\sqrt{k^2-1}}\lambda\sin(\lambda\tau),\qquad
a=\frac{\sqrt{k^2-1}}\lambda,\quad b=\lambda.}
$$
This is the <radial timelike geodesic in anti-de Sitter spacetime>. Its first return to the origin is
$$
\boxed{\tau_{\rm return}=\frac\pi\lambda,}
$$
independent of the outward launch speed for every nontrivial launch $k>1$. The coordinate $r$ is nonnegative, so this sine expression describes the excursion up to that return; subsequent passage through the origin requires changing the radial direction or using a signed diameter coordinate. The speed changes the amplitude, not the harmonic frequency.
\b[Circular periods.] The effective radial potential is $V_{\rm eff}=A(1+h^2/r^2)$. For a circle at $r_0$, $V_{\rm eff}'=0$ gives $h^2=\lambda^2r_0^4$, and normalization gives $k=A(r_0)$. Thus $\dot t=1$ and $\dot\phi=\pm\lambda$. A <stable circular timelike geodesic in anti-de Sitter spacetime> therefore has
$$
\boxed{\Delta t=\Delta\tau=\frac{2\pi}\lambda,}
$$
independently of radius. This is the same isochronous behavior as the central harmonic radial equation. A static clock placed at the orbit's radius instead measures $\sqrt{1+\lambda^2r_0^2}\,\Delta t$, which does depend on radius; it is a different clock from the orbiting particle's <proper time>.
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