A variation of conformal vector field

Question

Let $H$ be a smooth hypersurface in $\mathbb{R}^d$. I'm interested in vector field $X \in \mathfrak{X}(H)$ for which there is $f \in C^{\infty}(H)$ such that: \begin{align} \forall (u,v) \in \mathfrak{X}(H)^2, \quad L_Xg(u,v)=fII(u,v), \end{align} where $L$ is the Lie derivative, $g$ the induced metric on $H$ and $II$ it's second fundamental form. In the case of the euclidian sphere, which was my first interest, $II=g$ so that such vector fields are conformal. Is this relation well known in the general case? Can we say anything about $X$?

Answer 1

Using that

$$ L_X g = \mathrm{Sym}(\nabla X^\flat) $$

where $\nabla$ is the induced Levi-Civita connection and

$$ II(u,v) = - \langle D_u n,v\rangle_e $$

where $D$ is the Euclidean connection on $\mathbb{R}^d$, we get that your relation is equivalent to

$$ \mathrm{proj}_H \mathrm{Sym} ( D (X + fn)^{\flat_e}) = 0 $$

where $\flat_e$ is the lowering operation relative to the Euclidean ambient metric, and $\mathrm{proj}_H$ is the projection from the tensor bundle over $\mathbb{R}^d$ to that over $H$. Therefore a sufficient condition for $X$ is that that $X + fn$ is a Euclidean Killing vector field, and since $X$ is by definition tangential to $H$, this means that $X$ is the projection of a Euclidean Killing field to $H$. In the case of the sphere, this recovers the conformal fields of $\mathrm{S}^{d-1}$.

At the moment I am not sure whether there are other solutions, but am tempted to guess that there are not.