In a differential geometry paper from 1979 I have come across some terminology which I have not found explained anywhere else.

We have an immersion $x : S^2 \to S^n$. In the paper, it is a minimal immersion but I'm not sure it matters. It goes on to say

"Let $T_k(x)$ denote the real osculating space of order $k$ of $x$".

A) What is the precise definition of the real osculating space of an immersion in moden differential geometric language?

B) What does it mean intuitively?

[The paper is "An Extrinsic Rigidity Theorem for Minimal Immersions of S^2 into S^n" by J.L.M. Barbosa]

(I asked this on Stack Exchange originally)