Lovett S.T. Differential geometry of manifolds

2nd edition. — Boca Raton: CRC Press, 2019. — 450 p. — (Textbooks in Mathematics). — ISBN: 0367180464.The Second Edition presents the extension of differential geometry from curves and surfaces to manifolds in general. The book provides a broad introduction to the field of differentiable and Riemannian manifolds, tying together classical and modern formulations. The goal is to introduce manifolds in a both streamlined and mathematically rigorous way while keeping a view toward applications, particularly in physicsAnalysis of multivariable functions.
Variable frames.
Differentiable manifolds.
Multilinear algebra.
Analysis on manifolds.
Introduction to Riemannian geometry.
Applications of manifolds to physics.
Point set topology.
Calculus of variations.
Further topics in multilinear algebra.