I have a question that I would like to share to get some feedback.

In spectral methods, increasing the degree of a polynomial for a fixed element results in obtaining more points (for example, with a shape function of degree 2, we can accurately determine 3 points, and by increasing the degree, we can determine even more points, such as 20). However, we cannot infer information about the space between two points directly because, in Euclidean space, only a single straight line passes between two points. To gain information between points, we need to increase the number of elements or the degree of the shape function.

My question is: why don't we solve this problem in Riemannian space? Unlike Euclidean space, Riemannian space allows for curvature, meaning we do not have to rely on straight lines between two points. With this idea, we can obtain information between two points using lower-degree polynomials derived from Riemannian space.

My hypothesis is that the basis function created from Riemannian space inherently provides this feature, just as the basis function in Euclidean space inherently provides information between two points.

This is an idea that has come to my mind, and I would like to know your thoughts on how valid this idea might be.