taylor expansion on the unit sphere I am looking for a taylor type expansion for functions defined on the unit sphere $S^2$, is the following correct or what should be the right form:
$$f(y)-f(x)=\langle \nabla_g f(x), \gamma\rangle \Theta+higher\,\,order\,\, terms$$
where $\nabla_g$ is the gradient with respect to $g$,the the round metric, $\Theta$ is the angle which is the geodesic distance between $x$ and $y$, $\langle, \rangle$ is the Euclidean inner product and $\gamma$ is the direction between $x$ and $y$ which is tangent to the geodesic between $x$ and $y$ at the initial point $x$. Is there anything wrong? Please help, thanks a lot!

Note that the composition of the exponential map at x and f is a real valued function on the vector space $T_xS^2$. Hence you could use the classical Taylor expansion.
See also:
http://press.princeton.edu/chapters/absil/Absil_Chap7.pdf
