Examples of interesting / non-trivial manifolds that are direct products What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds.
Say, an $n$-torus $T^n$ is a direct product of $n$ copies of a circumference $S^1$. One can build a 3-manifold from a surface of genus $g$ as $M=M^2_g\times S^1$, and use somehow the connected sums of such manifolds.
Typically an "important" manifold would have a name or standard notation. For example, the Kodaira-Thurston manifold (important if it has a proper name!) decomposes into a product of the Heisenberg nil manifold and $S^1$.
I am looking for other important / interesting / non-trivial manifolds that happen to be direct products, preferably having important / interesting applications.

The Lie groups $U(n)$ are diffeomorphic to products $SU(n)\times S^1$ (but are not Lie isomorphic, unless $n=1$).
The Lie group SO(8) is diffeomorphic (but not Lie isomorphic) to $SO(7)\times S^7$.  (In fact, $S^7$ doesn't have a Lie group structure at all.)
The Lie group $SO(4)$ is diffeomorphic (but not Lie isomorphic) to $S^3\times \mathbb{R}P^3\cong S^3\times SO(3)$.
(To my knowledge, Lie groups with Lie algebra $\mathfrak{so}(8)$ are the only simple Lie group which are diffeomorphic to nontrivial products).
Relatedly, the unit tangent bundles to $S^1$, $S^3$, and $S^7$ are diffeomorphic to products $S^{k-1}\times S^k$, and no other unit tangent bundles of spheres are.  
