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If the two tangent spaces are equal, then the vectors in the two tangent spaces are exactly the same. I suspect that it's equivalent to picking a vector at ##p## and dragging it to ##q## and the vector will not change. So perhaps a test to check whether the tangent spaces are the same is to see if the covariant derivatives of any vector at ##p## w.r.t. a vector field which generates flow which maps the point ##q## vanish.

Is this a good test?