"Nice" metric on $\Bbb RP^3\#\Bbb RP^3$

Question

Is there a standard riemannian metric on $\Bbb RP^3\#\Bbb RP^3$, in view of geometrization conjecture? It is known that $\Bbb RP^3\#\Bbb RP^3$ has $S^2\times S^1$ as a double cover, so its universal cover is $S^2\times \Bbb R$. In this view, what would be a nice metric on this space?

Answer 1

The involution $\iota: S^2 \times S^1 \to S^2 \times S^1$ given by $\iota(v,z) = (-v, \overline z)$, the first component the antipodal map and the second component complex conjugation, is an oriented isometry, and so the standard metric on $S^2 \times S^1$ descends to the quotient.

Now prove the quotient is diffeomorphism to the connected sum of two projective 3-spaces. (Hint: consider the fundamental domain given by $S^2 \times I$, where $I$ is the north hemisphere of the unit circle.)