Is $VV^T + D$ a submanifold? If the positive definite matrix P forms a manifold,  is that the subset that {P: P = V V^T + D} where V is a low rank matrix  and D is a positive definite matrix a sub-manifold?  
This idea is corresponding to an application in finance that P is covariance matrix and V is a factor structure.
Thanks a lot.

What a coincidence. Someone asked a closely related (but different) question just two days ago.
Let $k\le n$ be a positive integer. Note that the five sets
\begin{align*}
\mathcal{P}&=\{P\in M_n(\mathbb{R}): P \text{ is positive definite}\},\\
\mathcal{S}_1&=\{D+VV^T: D\in\mathcal{P},\, V\in M_n(\mathbb{R}),\, \operatorname{rank} V\le k\},\\
\mathcal{S}_2&=\{D+VV^T: D\in\mathcal{P},\, V\in M_n(\mathbb{R}),\, \operatorname{rank} V=k\},\\
\mathcal{S}_3&=\{D+VV^T: D\in\mathcal{P},\, V\in M_{n,k}(\mathbb{R}),\, \operatorname{rank} V\le k\},\\
\mathcal{S}_4&=\{D+VV^T: D\in\mathcal{P},\, V\in M_{n,k}(\mathbb{R}),\, \operatorname{rank} V=k\},\\
\end{align*}
are identical to each other. So, whatever properties does $\mathcal{P}$ possess, those $\mathcal{S}_j$s possess the same properties too.
