Take away the mass.

Will the geometry exist?

Yes, it will. The reason why is because the equations of relativity dictate that:

[tex]\nabla_{\mu} R^{\mu \nu} = \frac{1}{2} \nabla_{\mu}g^{\mu \nu}R[/tex]

What this equation says is that if you remove all the mass in the universe, the curvature could still be non-zero. This is because of the presence, but highly ellusive today gravitational waves. So geometry can exist without matter.

It is for this reason, I believe that matter emerged from curvature. In effect, if you add enough curvature into a specific region, matter should appear.

Remember this is space-time, so if you really have no matter and energy for all space and all time, it turns out that the only thing that determines the shape of the universe is the cosmological constant. If you set that to zero, you get a flat universe everywhere. It still has geometry ("flat" is a geometry), but it has no curvature. Essentially, the entire universe has the geometry of special relativity. This is obviously a bit silly to take seriously, since what would a matterless/energyless universe mean?

Now, if you still take no traditional matter/energy, but add in a cosmological constant, then space can be curved, though you still have limited potential geometries.

If you're simply looking at a region of space that contains no matter/energy, then it can also be curved, since matter/energy elsewhere in the universe can be causing space-time to curve or it could be emitting gravitational waves.

I don't think anyone knows for sure whether having "enough curvature" could cause matter to appear spontaneously from the vacuum. It's intriguing, but probably needs quantum gravity, and we don't have a satisfactory theory for that yet.