How to find the normal vector of a point on a geodesic in a space described by a 3D metric tensor?

Question

I have the following metric tensor which varies in space. $$\widetilde{M}\left(\vec{r}\right)=\left[\begin{matrix}E\left(\vec{r}\right)&F\left(\vec{r}\right)&G\left(\vec{r}\right)\\F\left(\vec{r}\right)&H\left(\vec{r}\right)&I\left(\vec{r}\right)\\G\left(\vec{r}\right)&I\left(\vec{r}\right)&J\left(\vec{r}\right)\\\end{matrix}\right]$$ I also have an initial unit vector $\vec{T}$. There is a normal vector $\vec{N}$ which is perpendicular to $\vec{T}$. $$\begin{matrix}\vec{\gamma}\left(t\right)=Position\\\vec{T}\left(t\right)=\frac{\partial}{\partial t}\vec{\gamma}\left(t\right)\\\vec{N}\left(t\right)=\frac{\partial}{\partial t}\vec{T}\left(t\right)\\\end{matrix}$$ How would I go about finding the vector $\vec{N}$ which would correspond to that of a geodesic? I know that for a 2D surface in 3D space $\vec{N}$ is along the surface’s normal vector at that point. How do I solve this problem though in a 3D metric tensor though?