  In the past decades, the central limit theorem (CLT) has been generalized to
non-Euclidean data spaces. Some years ago, it was found that for some random
variables on the circle, the sample Fr\'echet mean fluctuates around the
population mean asymptotically at a scale $n^{-\tau}$ with exponent $\tau <
1/2$ with a non-normal distribution if the probability density at the antipodal
point of the mean is $\frac{1}{2\pi}$. The author and his collaborator recently
discovered that $\tau = 1/6$ for some random variables on higher dimensional
spheres. In this article we show that, even more surprisingly, the phenomenon
on spheres of higher dimension is qualitatively different from that on the
circle, as it depends purely on geometrical properties of the space, namely its
curvature, and not on the density at the antipodal point. This gives rise to
the new concept of geometrical smeariness. In consequence, the sphere can be
deformed, say, by removing a neighborhood of the antipodal point of the mean
and gluing a flat space there, with a smooth transition piece. This yields
smeariness on a manifold, which is diffeomorphic to Euclidean space. We give an
example family of random variables with 2-smeary mean, i.e. with $\tau = 1/6$,
whose range has a hole containing the cut locus of the mean. The hole size
exhibits a curse of dimensionality as it can increase with dimension,
converging to the whole hemisphere opposite a local Fr\'echet mean. We observe
smeariness in simulated landmark shapes on Kendall pre-shape space and in real
data of geomagnetic north pole positions on the two-dimensional sphere.
