Summary of comments and other sources
There are at least 4 similar concepts:
- Irreducible smooth manifold: As Ryan Budney said, "Regarding high dimensions, generally irreducible manifolds do not exist, this is because the connect-sum operation has some invertible objects -- in dimension 5 and up they are known as homotopy-spheres." See more in
Kosinski, Antoni A., Differential manifolds, Pure and Applied Mathematics, 138. Boston, MA: Academic Press. xvi, 248 p. (1993). ZBL0767.57001.
Kervaire, Michel A.; Milnor, John W., Groups of homotopy spheres. I, Ann. Math. (2) 77, 504-537 (1963). ZBL0115.40505.
Irreducible symmetric spaces: A symmetric space is
said to be irreducible if it is not isometric to a product of positive-dimensional
Irreducible vector space: A vector space is said to be irreducible if has no nontrivial invariant subspaces inside it.
and using case 3 we can talk about
- Irreducible Riemannian manifolds: Those manifolds where the holonomy has no invariant subspaces.