Connection between Gram matrix and Riemannian invariants? Recall that the Gram matrix of vectors $v_1, \dots, v_k\in\mathbb{R}^n$ is the $k\times k$ matrix $G_{ij}=(v_i,v_j)$. Now suppose that the vectors $v_i$ have been sampled uniformly from some submanifold $M\subset\mathbb{R}^n$. Is there some sensible notion of the limiting Gram matrix (as $k\to\infty$) in terms of the geometry of M? Maybe the largest eigenvalue converges to some Riemannian invariant of M? Any references on these kinds of questions would be appreciated!
 A: In the simplest case of vectors on the unit sphere in $\mathbb{R}^d,$ the off-diagonal entries of the Gram matrix will be Beta-distributed (with variance $1/d,$ see https://stats.stackexchange.com/questions/85916/distribution-of-scalar-products-of-two-random-unit-vectors-in-d-dimensions), so the matrix will not be the identity matrix. The eigenvalues will be the squares of the singular values of a random $d\times n$ matrix, for the distribution of which see the extensive oeuvre of Mark Rudelson and Roman Vershynin (see, for example, the ICM paper). It's fairly clear that if your random vectors are on an ellipsoid you can recover the semi-axes (and the center), and for a product of spheres the matrix will have a block structure.
What happens for a general submanifold, I have no clue. Notice that is not even clear how to express the expectation of a diagonal entry (which is the squared norm of a random point on your manifold) in terms of "usual" differential geometric (really "integral geometric" is what you are looking for here) invariants.
