Halos in non-standard analysis

Question

Please consider this question in terms of the hyperreals. As per usual, the halo of a point $P$ is the set of all points separated from $P$ by an infinitesimal distance. Let $P$ be a point in a curved 2D surface $\sigma$. Every point in the halo of $P$ is a point of $\sigma$ (meaning the halo is in $\sigma$.) Let the tangent plane to $\sigma$ at $P$ be $T_P$. My question is this: do all the points in the halo, which all lie in $\sigma$, also lie in the tangent space $T_p$?

Answer 1

No, in fact typically none of the points in the halo in $\sigma$ except for $P$ itself will be in $T_P$. For instance, if $\sigma$ is the unit sphere and $P=(1,0,0)$, then $T_P=\{(1,y,z):y,z\in{}^*\mathbb{R}\}$. But every other point in the halo of $P$ in $\sigma$ has $x$-coordinate strictly less than $1$ (though only infinitesimally so), and so is not in $T_P$.

More generally, if $\sigma$ is the nonstandard version of some standard surface $S\subset\mathbb{R}^3$ and $P\in S$, then if there is some neighborhood of $P$ in $S$ that does not intersect the tangent plane except at $P$, then the same is true of $\sigma$ by transfer, and so the halo will not intersect the tangent plane except at $P$.

Answer 2

This depends on how you define the tangent plane. According to one definition, the tangent plane is the quotient of suitable elements of the halo modulo an equivalence relation. This point of view was developed in detail in the following article:

Nowik, T; Katz, M. "Differential geometry via infinitesimal displacements." Journal of Logic and Analysis 7:5 (2015), 1-44.