Tangent Space to the Quadratic Form

Question

Let $S:=\{x \in \mathbb{R}^n : x^\top Ax = 1 \}$. We know that $S$ is a $(n-1)$-dimensional submanifold of $\mathbb{R}^n$ because it is a regular level set of a smooth function. What is the tangent space to $S$ at a given point $x_0$ (probably in terms of matrix $A$)? We can assume that $A$ is symmetric and invertible. Thank you, in advance, for your response!

Answer 1

(I'm using the Einstein summation notation, and that $A$ is symmetric.)

If you differentiate the function $f(x) = x^T A x = x_i A_{ij} x_j$, you get :

$f_k(x) = A_{kj} x_j + x_i A_{ik} = 2 A_{kj} x_j$

This lets you compute the gradient of $f$ , namely $\nabla f = 2 [ A_{1j} x_j, \ldots, A_{nj} x_j]$.

Now you can use the fact that the tangent space to a hypersurface is the hyperplane orthogonal to the gradient vector.

Discussion:

In particular, if you think of the variety as being defined by the restriction of the quadratic form $Q(v,w) = v^T A w$ to the diagonal $\{ (v,v) : v \in \mathbb{R}^n \} \cong \mathbb{R}^n$, and on the diagonal setting setting $Q(v,v) = 1$, then you can think of the tangent space to $Q(v,v)=1$ at $v$ as being defined by the linear form $Q(v,\_) = 0$.

Yet another point of view: we can think of $Q$ as defining an linear map $\tilde{Q} : \mathbb{R}^n \to (\mathbb{R}^n)^*$, namely $v \to Q(v, \_)$. Then the map that sends a point to it's normal (translated to the origin) is just the restriction of $\tilde{Q}$ to the set $Q(v,v) = 1$. (This is the Gauss map: https://en.wikipedia.org/wiki/Gauss_map )

Example:

A good example to think about all of this is the circle $x^2 + y^2 = 1$ and the sphere $x^2 + y^2 + z^2 = 1$. I'll leave it to you to work out the details in these cases, where you can check your computations against visual intuition.

Answer 2

If we find a chart $(W,G)$ for the $k$-dim submanifold $S$ at point $x_0$, we can describe the tangent space as $$T_{x_0}S=\{x_0\} \times \{ v \in \mathbb{R}^n: v=DG(G^{-1}(x_0))\xi, \xi \in \mathbb{R}^k \}.$$ For example, if we consider the submanifold $x^2 + y^2 = 1$, and assume that $(x_0,y_0) \in \{(x,y): y=+\sqrt{1-x^2} \}$, then we have: $$T_{x_0}S=\{(x_0,y_0)\} \times \{ v \in \mathbb{R}^2: v=(\xi,\frac{-x_0}{\sqrt{1-x_0^2}} \xi), \xi \in \mathbb{R} \}. $$ Similarly, we can describe the tangent space for other points on this manifold. My question is that: How can we generalize this? i.e., what is a local chart for the submanifold $x^\top A x = 1$ ?