I thought the only 3-manifold with positive constant curvature is $\mathbb{S}^3$. But I faced $\mathbb{S}^3/\Gamma$, where $\Gamma$ is a finite subgroup of $SO(4)$ and surprised! My problem is that I can not imagine any other manifolds which have positive constant curvature, except of $\mathbb{S}^3$! I have no idea how mathematicians have discovered this concept! It's great for me! Of course I am wondered because of having little information.
Can anyone help me to visualize $\mathbb{S}^3/\Gamma$? And explain to me how can I show the curvature of $\mathbb{S}^3/\Gamma$ is constant?
Thanks
If $(M, g)$ is a Riemannian manifold and $\Gamma$ is a discrete (e.g., finite) group of isometries of $(M, g)$ acting without fixed points, then the quotient space $M/\Gamma$ (which is a manifold because $\Gamma$ acts without fixed points) admits a unique Riemannian metric with respect to which the quotient map is a local isometry. This metric obviously inherits local properties of $g$, such as having constant (scalar, Ricci, sectional) curvature.
Your ability to visualize quotients of $S^3$ by a finite group is likely to be as much topological as geometric. Presumably you've seen the Wikipedia page on spherical manifolds?
The simplest non-trivial example of your setup occurs when $\Gamma$ is the multiplicative group $\{1, -1\}$, acting by scalar multiplication on $\mathbf{R}^4 \simeq \mathbf{C}^2$. The Riemannian quotient is a constant-curvature metric on the projective space $\mathbf{RP}^3$ (which is diffeomorphic to the rotation group $SO(3)$, a.k.a., the positive orthonormal frame bundle of the 2-sphere $S^2$, if that's helpful).
Other cyclic groups acting on $S^3$ give rise to lens spaces. Another famous and important quotient is the Poincare homology sphere (MO discussion; see also the Wikipedia link above).
You may be able to glean a certain amount of geometric intuition by imagining a fundamental domain equipped with a metric of constant curvature and picturing the necessary boundary identifications. The "usual mental kludge" is to imagine "inflating" a solid region $D$ in $\mathbf{R}^3$ in such a way that the boundary identifications geometrically "close up" along the edges and vertices of $D$, i.e., a total angle of $2\pi$ is incident in each normal plane to each edge of $D$, and $4\pi$ steradians are incident at each vertex.