Many of us across diverse application domains--data mining, handwriting recognition, computer vision, medical image analysis--are finding that we need at least a passing familiarity with differential geometry. Differential geometry supplies tools to do computations and analysis in nonlinear spaces or manifolds.
In medical imaging, the nonlinear manifolds might be the shape spaces of continuous curves or surfaces, or it might be the space of positive definite matrices used in diffusion tensor imaging (DTI). In computational geometry, it might be the space of 3D rotations SO(3) and in data mining, the working assumption is that the ubiquitous high dimensional data actually reside in tractable lower dimensional spaces--in nonlinear manifolds.
Some of these useful tools from differential geometry are tangent spaces, geodesics, exponential maps and inverse exponential or log maps.