Diffusion tensor imaging (DTI) has become the major modality to study properties of white matter and the geometry of fiber tracts of the human brain. Clinical studies mostly focus on regional statistics of fractional anisotropy (FA) and mean diffusivity (MD) derived from tensors. Existing analysis techniques do not sufficiently take into account that the measurements are tensors, and thus require proper interpolation and statistics based on tensors, and that regions of interest are fiber tracts with complex spatial geometry. We propose a new framework for quantitative tract-oriented DTI analysis that includes tensor interpolation and averaging, using nonlinear Riemannian symmetric space. As a result, tracts of interest are represented by the geometry of the medial spine attributed with tensor statistics calculated within cross-sections. Examples from a clinical neuroimaging study of the early developing brain illustrate the potential of this new method to assess white matter fiber maturation and integrity.