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Is there way to represent the collection of all smooth/continuous functions invariant under the antipodal map on the surface of the n-Sphere in the n+1 dimensional real space?

In other words, is there a finite set of smooth continuous functions, invariant under antipodal map, that can represent paths on an n sphere?

I have that for the 2-sphere, and a point p on the sphere, any pair of deg 2 monomials in 3 coordinates i.e. $x^2, y^2, z^2, xy, yz, zx$ can be used to represent smooth neighborhoods of the point p. For instance $(x^2, y^2)\text{ or }(xy, yz)$ can be thought of as arbitrary neighborhoods of p. Transition functions on the intersection of these neighborhoods is a path on the sphere!

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The answer to this question is in Spherical Harmonics and representation theory. arXiv has a very good intro to the topic and describes smooth functions on the n-sphere.

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