Definition: An immersion $f:M^{n}\rightarrow \mathbb{R}^{m}$ is said to envelop the congruence of hyperspheres $S(h(x),r(x))$ whose centers and radii are given by $h:M^{n}\rightarrow \mathbb{R}^{m}$ and $r \in C^{\infty}(M)$ if $$ f(x) \in S(h(x), r(x)) \quad \mbox{and}\quad f_{*}T_{x}M \subset T_{f(x)}S(h(x), r(x))$$ that is, if $$||f(x)-h(x)||^{2}=r^{2}(x)\quad \mbox{and} \quad \langle f_{*}X, f(x)-h(x) \rangle=0$$ for all $x \in M^{n}$ and $X \in T_{x}M$.
Assume the following result that is proved by using Lorentz differential geometry:
Theorem: Let $f, \tilde{f}:M^{n}\rightarrow \mathbb{R}^{n+1}$ be immersions that envelop a common congruence of $n-\mbox{dimensional}$ spheres in $\mathbb{R}^{n+1}$. Besides, Let $f,\tilde{f}$ be conformal immersions, that is, $$ \langle, \rangle_{\tilde{f}}=\varphi^{2}\langle, \rangle_{f}$$ where $\varphi \in C^{\infty}(M)$ is positive. Then there exists $\eta \in \Gamma(T^{\perp}_{f}M)$, $Z \in \Gamma(TM)$ and $b \in C^{\infty}(M)$, satisfying $$ \alpha^{f}(X,Z)-\langle X, Z\rangle_{f} \eta-b\nabla^{\perp}_{X}\eta=0$$ for all $X \in \Gamma(TM)$, such that $$ \tilde{f}(x)=f(x)+\varphi(f_{*}Z-b\eta)$$ for all $x \in M^{n}$, where $\varphi=\dfrac{-2b}{||f_{*}Z-b\eta||^{2}}$. Here $\alpha^{f}$ is the second fundamental form of $f$.
My question: I would like to know if someone could show me that the vector field $Z$ is a gradient vector field, that is, there exists $\rho \in C^{\infty}(M)$ such that $$ Z=\mbox{grad}\,\rho,$$ or equivalently, show that $$ \langle \nabla_{X}Z,Y\rangle_{f}=\langle \nabla_{Y}Z,X \rangle_{f}$$ where $\nabla$ is the Levi-Civita connection with respect to the metric induced by $f$.
Thak you for your help.