In this talk, we discuss three problems in geometric data analysis. The two first problems are concerned with dimensionality reduction, on one hand for shapes in the sense of Kendall, and on the other hand for network-valued data. Such objects are modeled as elements of a quotient space equipped with a Riemannian structure. In order to construct lower-dimensional representations within this geometric framework, we investigate the key notion of exponential barycenter. In the last problem, we come back to statistical shape analysis, specifically to the analysis of n-dimensional curves. The problem is one in single-cell data analysis known as trajectory inference, namely that of reconstructing a differentiation tree from sequencing data. To this end, we study the property of varifold distances to characterize the topology of a vector field — in our case a RNA velocity field — based on its integral curves.