This textbook offers an introduction to differential geometry designed for readers interested
in modern geometry processing. Working from basic undergraduate prerequisites the authors
develop manifold theory and Lie groups from scratch fundamental topics in Riemannian geometry
follow culminating in the theory that underpins manifold optimization techniques. Students and
professionals working in computer vision robotics and machine learning will appreciate this
pathway into the mathematical concepts behind many modern applications.Starting with the matrix
exponential the text begins with an introduction to Lie groups and group actions. Manifolds
tangent spaces and cotangent spaces follow a chapter on the construction of manifolds from
gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector
fields and basic point-set topology bridge into the second part of the book which focuses on
Riemannian geometry.Chapters on Riemannian manifolds encompass Riemannian metrics geodesics
and curvature. Topics that follow include submersions curvature on Lie groups and the
Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds
and symmetric spaces revealing the machinery needed to generalize important optimization
techniques to Riemannian manifolds. Exercises are included throughout along with optional
sections that delve into more theoretical topics.Differential Geometry and Lie Groups: A
Computational Perspective offers a uniquely accessible perspective on differential geometry for
those interested in the theory behind modern computing applications. Equally suited to
classroom use or independent study the text will appeal to students and professionals alike
only a background in calculus and linear algebra is assumed. Readers looking to continue on to
more advanced topics will appreciate the authors' companion volume Differential Geometry and
Lie Groups: A Second Course.
