Given a smooth scalar function $f(x_1, x_2, \ldots, x_n)$ defined in $\mathbb{R}^n$, we have at each point a gradient vector $g=\nabla f$ and a Hessian matrix $H$ with $H_{ij}=\partial _{x_i}\partial_{x_j}f$.

Suppose $C(t)$ $(t\ge0)$ is an integral curve of the gradient vector field $g$, with the property that at the starting point $t=0$ the tangent vector of $C$ is an eigenvector of the Hessian matrix at that point.

My question is, is it true that for $t>0$ the tangent vector of $C(t)$ keeps being an eigenvector of the Hessian matrix at the corresponding positions?