If you have data that are sampled from a normal distribution, what is the relationship between the arithmetic and geometric means? Would it ever make sense to report the geometric mean instead of the arithmetic mean? (Assume that all the values are positive; no zeros, no negative values)
Why do you want a geometric mean of normal distributed observations? I can see no good reason. Applications of the Geometric Mean gives several good examples of use of geometric mean. A typical case is return on investment. Returns combine multiplicatively, so If you want one "typical" return that would result in same winning, if the return was held constant over years, you get that from the geometric mean of the yearly returns.
Common to all such examples is that the random variable in question cannot be negative, and, since every normal distributed variable is negative with some (maybe very small) positive probability, geometric means do not look natural to use. So, again, why do you want to use a geometric mean?
Edited because I am an idiot:
If you have normal samples you might have negatives depending on your mean and variance which would flip the sign. It feels unstable to flip a sign irrespective of the magnitude. Bummer.
More interesting are distributions with support on the positive or negative reals.