Roughly speaking, the type of objects I'm thinking of are something like half-infinite strips, $$\{(x,y)\in \mathbb{R}^2:x\in [0,1],y\in [0,\infty)\},$$ and my question concerns the behavior at the point (0,0). The question can be summed up as the following: on an orientable manifold-with-boundary $M$, are normal vectors defined for a codimesion 2 submanifold $N\subset M$, where $N\subset \partial M$? More generally, does it make sense to talk about $T_p M$ where $p\in N$?
To start, define the standard square pyramid $P$, consider the unit square $$\{(x,y,0):x,y\in [0,1]\},$$ and consider the point $p=(\frac{1}{2},\frac{1}{2},1)$. We define $P$ as all convex combinations of $p$ and $S$. We say the interior of the base face is the set $$\{(x,y,0):x,y\in (0,1)\},$$ and the boundary of the base face is $\{(x,y,0):x+y=1\}$.
We say a Riemannian manifold $(M,g)$ together with a Lipschitz diffeomorphism $\phi:M\to P$, where $P$ is a standard pyramid is a square pyramid type if $\phi^{-1}$ restricted to the interior, faces and edges of $P$ is smooth. Call the interior and boundary of base face of $B\subset \partial M$ the image under $\phi^{-1}$ of the corresponding sets of the standard pyramid.
Consider the outward unit normal vector field on $B$. Clearly this vector field is well-defined in the interior of $B$. Is there a sense in which can uniquely extend this vector field to the boundary of $B$?
Edit: For clarity, I would be interested in understanding the question for the half-infinite strip, $$M=\{(x,y)\in \mathbb{R}^2:x\in [0,1],y\in (0,\infty)\}.$$ In this case, $\partial M=[0,1]\times \{0\}$ and $N=(0,0)\sqcup (1,0)$. In this case, we can define the define outward unit normal on $\partial M$. Can we extend the unit outward normal to $N$?