We used tensor-derived measures to map the extent of genetic effects on brain fiber microstructure, in 23 monozygotic and 23 dizygotic twin pairs. All 92 DTI volumes were fluidly registered to a geometrically-centered template via a high-dimensional mapping of co-registered structural-MRI. After tensor re-orientation, we computed three scalar DTI measures: the fractional anisotropy (FA), geodesic anisotropy (GA), and the hyperbolic tangent of GA (tGA); GA measures the geodesic distance between tensors on the symmetric positive-definite tensor manifold. Spatial maps of intraclass correlations between MZ and DZ twins were compared to compute maps of Falconer’s heritability statistics. We also performed a maximum likelihood estimation of genetic influences using path analysis. The manifold-based measure, tGA, was marginally more powerful than FA for detecting genetic influences, and improved the fit of quantitative genetic models relative to FA and GA. The pattern of genetic influences was remarkably consistent with the neurodevelopmental sequence, with strong occipital genetic effects and strong frontal environmental effects.