Q:

Geometry of the curvature of spacetime around rotating or spinning revolving objects

What is the geometry of the curvature of spacetime around rotating or spinning revolving objects, or anything that has an angular momentum?

A:

Take a small object, such as a  golf ball,  it will have an undetectable spacetime distortion,  from its own mass, nor from any angular momentum.
So leave the mass fixed, and spin the golfball  up to 99.99.999999..... of light speed, assuming it will not explode long before that. So there will still be no GR mass induced spacetime distortion, but Frame Dragging will occur.

Rotational frame-dragging (the Lense–Thirring effect) appears in the general principle of relativity and similar theories in the vicinity of rotating massive objects. Under the Lense–Thirring effect, the frame of reference in which a clock ticks the fastest is one which is revolving around the object as viewed by a distant observer. This also means that light traveling in the direction of rotation of the object will move past the massive object faster than light moving against the rotation, as seen by a distant observer. It is now the best known frame-dragging effect, partly thanks to the Gravity Probe B experiment. Qualitatively, frame-dragging can be viewed as the gravitational analog of electromagnetic induction.

The Gravity B probe  is well covered on Wikipedia. 

What is the geometry of the curvature of spacetime around rotating or spinning revolving objects or anything that is having an angular momentum

Frame-dragging and it's effects on spacetime geometry  are best described by the  Kerr metric, taking a mass $M$ and angular momentum $J$
$${\displaystyle {\begin{aligned}c^{2}d\tau ^{2}=&\left(1-{\frac {r_{s}r}{\rho ^{2}}}\right)c^{2}dt^{2}-{\frac {\rho ^{2}}{\Lambda ^{2}}}dr^{2}-\rho ^{2}d\theta ^{2}\\&{}-\left(r^{2}+\alpha ^{2}+{\frac {r_{s}r\alpha ^{2}}{\rho ^{2}}}\sin ^{2}\theta \right)\sin ^{2}\theta \ d\phi ^{2}+{\frac {2r_{s}r\alpha c\sin ^{2}\theta }{\rho ^{2}}}d\phi dt\end{aligned}}}$$
 $r_s$ is the Schwarzschild radius
${\displaystyle r_{s}={\frac {2GM}{c^{2}}}}$
${\displaystyle \alpha ={\frac {J}{Mc}}}$
${\displaystyle \rho ^{2}=r^{2}+\alpha ^{2}\cos ^{2}\theta \,\!}$
${\displaystyle \Lambda ^{2}=r^{2}-r_{s}r+\alpha ^{2}\,\!}$

In the non-relativistic limit where $M$ (or, equivalently, $r_s$) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates

$${\displaystyle c^{2}d\tau ^{2}=c^{2}dt^{2}-{\frac {\rho ^{2}}{r^{2}+\alpha ^{2}}}dr^{2}-\rho ^{2}d\theta ^{2}-\left(r^{2}+\alpha ^{2}\right)\sin ^{2}\theta d\phi ^{2}}$$

We may rewrite the Kerr metric in the following form

$${\displaystyle c^{2}d\tau ^{2}=\left(g_{tt}-{\frac {g_{t\phi }^{2}}{g_{\phi \phi }}}\right)dt^{2}+g_{rr}dr^{2}+g_{\theta \theta }d\theta ^{2}+g_{\phi \phi }\left(d\phi +{\frac {g_{t\phi }}{g_{\phi \phi }}}dt\right)^{2}}$$

This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radiusr and the colatitude θ

$${\displaystyle \Omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {r_{s}\alpha rc}{\rho ^{2}\left(r^{2}+\alpha ^{2}\right)+r_{s}\alpha ^{2}r\sin ^{2}\theta }}}$$

In the plane of the equator this simplifies to

${\displaystyle \Omega ={\frac {r_{s}\alpha c}{r^{3}+\alpha ^{2}r+r_{s}\alpha ^{2}}}}$

Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging.

