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This question refers to the last part of Chapter 1, Volume 2, of Spivak's Comprehensive Introduction to Differential Geometry (roughly corresponding to pp. 36-48).

It is known that, for planar curves, the only curves of constant affine curvature, up to a transformation in $SL(2,\mathbb{R})$, are the conic sections.

What about curves of constant affine curvature in $\mathbb{R}^n$?

I believe this corresponds to paths $[a,b] \to SL(n,\mathbb{R})$ but I am not 100% certain.

A pointer to a reference will suffice for an answer.

My guess (attempting to generalize from the case $n=2$) would be that they correspond either to: (1) conic sections embedded in $\mathbb{R}^n$, (2) curves defined by polynomials of degree $\le n$.

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    I imagine that you have add a look to (https://en.wikipedia.org/wiki/Affine_curvature). – 2017-02-02
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    @JeanMarie Yes, you are right. That article only mentions plane curves. This article: https://en.wikipedia.org/wiki/Affine_geometry_of_curves mentions curves in arbitrary $\mathbb{R}^n$, but doesn't state the higher-dimensional analog of the result for $n=2$ (assuming that such an analog even exists -- I don't know). – 2017-02-02

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