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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)

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View the immersion $x$ as an immersion into $R^{n+1}$. Then for each $p \in S^2$, there is a unique polynomial map $O_k: R^2 \rightarrow R^{n+1}$ of degree $k$ such that $O_k(0) = x(p)$ and the partials of $O_k$ of order $k$ or less at $0$ are equal to the corresponding covariant derivatives of $x$ at $p$.

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Sorry - so do you know how the osculating space is then defined using this polynomial map? – Spencer Jan 25 '13 at 10:38
It would be the image of the osculating map. – Deane Yang Jan 25 '13 at 14:19

I took a quick look around and Wolfram Mathworld has a great animation of an osculating circle here. Also the entry on osculating curves here has a nice definition. The intuition is that when the you look at $S^{2}$ immersed in $S^{n}$ you take a point $x$ and ask for another surface which also contains $x$ and both the immersion of $S^{2}$ and this new surface have the same $k$ derivatives at $x$. So a tangent plane is an osculating surface of order 1.

However perhaps the best source of intuition is what the word osculate means in latin, ``to kiss.''

As for what the modern algebraic geometry term might be, I did some quick Googling for ``osculating algebraic geometry" and the term still seems to be in use. Perhaps not very commonly.

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