In this paper we develop a novel approach for the scale-space representations of scalar functions defined over Riemannian manifolds. One of the main interest in such representations stems from the task of 3D modeling where 2D surfaces, endowed with various physical properties, are recovered from images. Multi-scale analysis allows to structure the information with respect to its intrinsic scale, hence enabling a wide range of low-level computations, similar to what is usually used for representing images. In contrast to the Euclidean image domain, where scale spaces can be easily obtained through convolutions with Gaussian kernels, surfaces require a more general approach that must handle non-Euclidean spaces. Such a generalized scale-space framework is the main contribution of this paper, which builds on the spectral decomposition available with the heat-diffusion framework to derive a computational approach for representing scalar functions on 2D Riemannian manifolds using an intrinsic scale parameter. In addition, we propose a feature detector and a region descriptor, based on these representations, extending the widely used DOG detector and HOG descriptor to manifolds. Experiments on real datasets with various physical properties, i.e., scalar functions, demonstrate the validity and the interest of this approach.