What is the Riemannian geometry's equivalent definition of ``Simply connectedness"? Sorry if my question is ridiculous or meaningless. I want to know how researcher uses from simply connectedness in their calculations in Riemannian geometry and not in the language of (de Rham) cohomology. i.e. 

What is the translation of topological notion Simply connectedness in Riemannian geometry?

 A: Simply contentedness is used in Riemannian geometry exactly as it is used in topology. Riemannian manifolds are still topological spaces, after all, ant the interplay of topology and geometry is a recurring theme. That said, there is an equivalent definition which might have a more "Riemannian" flavor.
Recall that every connected manifold $M$ has a unique simply connected covering manifold, the universal cover $\widetilde{M}$. Furthermore, the fundamental group $\pi_1(M)$ acts on $\widetilde{M}$ such that $M$ is equal to the quotient $\widetilde{M}/\pi_1(M)$. Note also that the covering map $\pi:\widetilde{M}\to M$ is a local diffoemorphism.
If $M$ is Riemannian with metric $g$, we can equip $\widetilde{M}$ with the pullback matric $\pi^*g$. This Riemannian universal cover has lots of nice properties. Namely, $\pi$ becomes a local isometry, geodesics lift onto geodesics, and $\pi_1(M)$ acts by isometries on $\widetilde{M}$. A simply connected manifold is then one which is equal to its Riemannian universal cover.
As an example of the usefulness of this construction, a complete, simply connected manifold with constant sectional curvature is uniquely determined (up to isometry) by its dimension and the value of its sectional curvature. This is a key result in the study of space forms, since it classifies their universal covers.
