I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical coordinates. I first want to investigate a metric for the Euclidean case, i.e. "flat" space-time with no matter or energy present. How would I write this Euclidean metric using a higher-dimensional analogue of spherical coordinates?
Aren't you basically asking to the formula of higher dimensional vectors to compute yourself being in the sphere at the center with appropriate lines to all vector points?
I think I get what your asking for I asked the same questions when I was younger
Anyways here's a hyperlink to 6th dimensional operation. The operator is used in quantum physics, but clearly they are talking about bosons and fermions. The reason I am referencing this is because this goes in the quantum level from one to six dimensions as far as the x y and the z but they use all this jargon now referencing your sphere of influence that you were talking about you could always plug in the sphere formula into the equation