Hi, I'm a newbie here and I would like to kindly ask for your collective wisdom on this forum. I am working on a computational technique to describe a time series data on U(2), for now. Given two points x and y in U(2), I can take a matrix log of x'y to find the tangent emanating from x to y. Using the one-parameter exponential map, I can find the points along the geodesic using the so obtained tangent. My thought was to have a basis representation of the tangent space so as to come up with quantized approximate tangent which can best (in terms of geodesic distance) describe the computed tangent. Intuitively, I want to form a grid of tangent directions which could be used at any point on U(2). Does there exist such a basis? Where can I look for such formulations? I apologize in advance if this a naive thought as I have no formal training in Lie group/Lie algebra. Thanks in advance for any help.