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I would guess that the answer is `not really'. As far as I know there is not a even a universally accepted definition of the 'normal distribution' on a Remanian Manifold. Probably the closest thing to the normal are those distributions that arise from generalisations of Brownian motion on manifolds. math.northwestern.edu/~ehsu/…
As I said the $x_i^2$ gets in the way. If is was just two multinomials, one to power $k$, the other to power $m$, there would be no problem, you would get $\exp(tx + sx)$ and everything would work out nicely as before. Perhaps I have missed something though. How do you intend to use the integral of $\exp(tx^2 + sx)$ (which has no closed form solution as far as I am aware) taken to the power of $n$ to efficiently compute the answer?
I like the updated part of this question +1. I recommend deleting the first part (However, don't do this if you have some specific reason not too). If you don't mind me asking, in what application does this problem occur? Also, what justifies your use of the von Mises Fisher distribution here?
