To me, in the context of data analysis, it is always linked with the idea of leaning a topographic mapping of the data, so that samples that are mapped close by, are similar in a given sense. The wikipedia site on Nonlinear dimensionality reduction offers a nice overview. THe paper Laplacian eigenmaps and Spectral Techniques for Embedding and Clustering contains a nice description of a framework where the idea of manifold learning is linked with differential geometry.
In other words, curvilinear is to me related to the problem of learning a distance metric from data. The hypothesis is that the data lies in a smooth, low dimensional manifold. That learned metric correspond to the metric tensor as in the classical sense of the term.