Note. My question is based on Lee's Introduction to Riemannian Manifolds, exercise 8-1.
Let $f: \mathbb{R}^n \to \mathbb{R}$ be a smooth function. Its graph
$$ S := \{ (x, f(x)) : x \in \mathbb{R}^n \} $$
is a hypersurface in $\mathbb{R}^{n+1}$. It has a global parametrization $x \mapsto (x, f(x))$, with the corresponding coordinates $(x^1, \dots, x^{n+1})$ on $S$. I would like to know the shape operator in these coordinates.
My attempt so far:
Define $F: \mathbb{R}^{n} \times \mathbb{R} \to \mathbb{R}$ by $F(x, y) := f(x) - y$ is a defining map of $S$, i.e. $S = F^{-1}(0)$. The unit normal field on $S$ is given by
$$ N = \frac{\text{grad }F}{\lVert \text{grad }F \rVert} \, , $$
where
$$ \text{grad }F = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \frac{\partial}{\partial x^i} - \frac{\partial}{\partial x^{n+1}} \, , $$
with length
$$ \lVert \text{grad }F \rVert = \sqrt{\sum_{i=1}^n \left(\frac{\partial f}{\partial x^i}\right)^2 + 1} \, . $$
Now compute the shape operator: For any vector field $\mathfrak{X}(S) \ni X = \sum_{j=1}^{n+1} X^j \frac{\partial}{\partial x^j}$
$$ \begin{align} sX &= -\bar{\nabla}_X N \\[5pt] &= - \frac{1}{\lVert \text{grad }F \rVert} \left( \sum_{i=1}^n \sum_{j=1}^{n+1} X^j \left( \frac{\partial}{\partial x^j} \frac{\partial f}{\partial x^i} \right) \frac{\partial}{\partial x^i} + \sum_{j=1}^{n+1} X^j \left( \frac{\partial}{\partial x^j} 1 \right) \frac{\partial}{\partial x^{n+1}} \right) \\[5pt] &= - \frac{1}{\lVert \text{grad }F \rVert} \sum_{i=1}^n \sum_{j=1}^{n+1} X^j \frac{\partial^2 f}{\partial x^i \partial x^j} \frac{\partial}{\partial x^i} \end{align} $$
My questions:
- First of all, would the above computation be correct?
- Can I write $\sum_{i=1}^n \frac{\partial f}{\partial x^i} \frac{\partial}{\partial x^i}$ as $\text{grad }f$? I'm confused as the coordinate is defined on $\mathbb{R}^{n+1}$, while $f$ is a function on $\mathbb{R}^n$, thus $\text{grad }f \in \mathfrak{X}(\mathbb{R}^n)$.
- Likewise, if in addition $Y \in \mathfrak{X}(S)$, can I write $\langle sX, Y \rangle = -\frac{1}{\lVert \text{grad }F \rVert} \text{Hess }f(X, Y)$? I'm unsure as $\text{Hess }f$ is a covariant 2-tensor field in $\mathbb{R}^n$, but $X, Y \in \mathfrak{X}(S) \subseteq \mathfrak{X}(\mathbb{R}^{n+1})$.