I am a beginner to Riemannian geometry. Following is my question.
In the Euclidean space, say $\mathbb{R}^3$, let us consider a plane, for simplicity, say one passing through the origin, $\mathbb{P}=\{\textbf{x}=(x_1,x_2,x_3):ax_1+bx_2+cx_3=0\}$ and consider the origin $\textbf{0}=(0,0,0)\in \mathbb{P}$.
Now, let us consider the set of all points ($\mathbb{L}$) in $\mathbb{R}^3$ whose best approximation with respect to $\ell_2$-distance on $\mathbb{P}$ is $\textbf{0}$, i.e., $$\{\textbf{y}:\arg \min_{\textbf{x}\in \mathbb{P}}\|\textbf{x}-\textbf{y}\|_2=\textbf{0}\}$$ which is precisely the line $$\mathbb{L}:~~~~~~~\frac{x_1}{a}=\frac{x_2}{b}=\frac{x_3}{c}$$
This line $\mathbb{L}$ can also be characterised by $\mathbb{L}=\{\textbf{y}:\langle \textbf{x},\textbf{y}\rangle=0 \text{ for all } \textbf{x}\in \mathbb{P}\}$.
My question is, how to define a Riemannian metric $g$ with components $g_{i,j}$ and a connection $\nabla$ with connection coefficients $\Gamma_{i,j}^k$ so that $\mathbb{L}$ is a $\nabla$-geodesic and prove that it intersects the plane $\mathbb{P}$ at the single point $\textbf{0}$?
Thank you.