After using the Log map, as defined in this paper Riemannian approaches in Brain-Computer Interfaces: a review (Section III. A page 2&3), to project points from the manifold onto the tangent space $T_P(M)$, do we use the metric defined by the ambient Euclidean space or the Riemannian metric to measure distances and norms of the vectors in the tangent space?
On one hand, I feel that the Riemannian metric should be used since it is what defines angles, distances, and norms on the tangent space. However, on the other hand, the tangent space $T_P(M)$ obtained through the Log map seems to be a Euclidean space, which can be viewed as a local linearization of the manifold around the point P (which I think can be viewed as a chart)?
I think this might be related to pullback, induced metrics, and these posts What is the difference between a chart and a tangent space? and Tangent Space and Charts, but I do not have a background in this field at all, so if someone could point me in the right direction and help clear up terminology and confusion that would be great! If the answer is the Euclidean metric, could someone explain why after the log map we no longer use the Riemannian metric to compare vectors in the tangent space. Alternatively, if the answer is the Riemannian metric, could someone explain why the projection onto the tangent space using the log map is not a local linearization and the geometry/curvature induced by the metric is not already accounted for by the log map?
Edit: My best guess answer based on intuition
If you project a set of points via the log map onto the tangent space centered at their mean, you have created a local linearization of the manifold centered around the mean. If you are not interested in the geodesic distance between these points but are just interested in clustering or finding linear distances between points then I believe using the metric defined by the ambient Euclidean space should suffice. Since the log map approximates the manifolds geometry close points on the manifold will be close in the tangent space and far points will be far (given your approximation is accurate).
However, since I believe the using the Riemannian metric on the tangent space allows us to measure how far movement in the tangent space corresponds with movement on the manifold, if you are interested in approximating the geodesic distance between projected points then you should use the Riemannian metric. Additionally, I believe if you are interested in comparing projected points on the tangent space to any other points on the tangent space (not necessarily from the set you projected) then you should use the Riemannian metric as well since this will again give you an estimate as to how far on the manifold these points actually are.