Direct product of spheres with constraint

Question

Suppose we have a space which is a direct product of 1-spheres: $\mathbb{S}^1\times\mathbb{S}^1\times...\times\mathbb{S}^1 = \mathbb{T}^N$ (the total number of spheres = $N$), or 2-spheres: $\mathbb{S}^2\times\mathbb{S}^2\times...\times\mathbb{S}^2$. The $\theta_i$ is the polar angle of the point on the sphere number $i$. Let we have also a constraint: $\sum\limits_{i=1}^N \cos(\theta_i) = const$, and $(-N)<const<N$. This constrain significantly changes this topological spaces and the question is how to identify the resulting space. May be some isomorphism to something conventional is possible ?