Is there a name for the manifold of centered, unit norm vectors?

Question

I am interested in solving a manifold optimisation problem where the manifold is described by: $\{ {\bf x}\in\mathbb{R}^n : {\bf x}^T{\bf x}=1 \land {\bf x}^T{\bf 1}=0 \}$. i.e. the set of centered vectors with unit 2-norm. The unit 2-norm constraint makes it like the oblique manifold but I'm wondering if there is a name for the manifold with both constraints?

Answer 1

You seem to have misunderstood what the "oblique manifold" is, at least according to this source. Until today, I hadn't heard of the term "oblique manifold".

In any case, a notable thing about the Euclidean sphere $$ S^{n-1} = \{x \in \Bbb R^n : x^Tx = 1\} $$ is that it is invariant under orthogonal changes of basis. A consequence, as it turns out, is that $\{x \in \Bbb R^n : x^Tx = 1 \wedge v^Tx = 0\}$ will be "the same manifold", no matter which non-zero $v$ is chosen.

With that in mind, taking $v = e_n = (0,\dots,0,1)$ gives us the set $$ \{x \in \Bbb R^n : x^Tx = 1 \wedge e_n^Tx = 0\} = \{(x_1,\dots,x_{n-1},0): x_i \in \Bbb R, x_1^2 + \cdots + x_{n-1}^2 = 1\} $$ In other words: we may regard the above set as an embedding of $S^{n-2}$ in the hyperplane orthogonal to $e_n$. Similarly, your set is an embedding of $S^{n-2}$ in the hyperplane orthogonal to $\mathbf 1$.