The volume develops the theory of smooth manifolds, tensor fields, differential forms, affine and Levi-Civita connections, geodesics, exponential maps, curvature, and pseudo-Riemannian geometry. These tools are introduced in a concise but systematic way, combining abstract geometric concepts with explicit coordinate expressions and applications relevant to physics.

After establishing the basic language of differential geometry, the text turns to the geometric structure of spacetime. It discusses Lorentzian manifolds, causal vectors and curves, time orientation, proper time, reference frames, the equivalence principle, conservation laws, Killing vector fields, Fermi-Walker transport, geodesic deviation, and the role of curvature in gravitation. The exposition then leads naturally to Einstein's field equations, their geometric meaning, and selected applications in relativistic physics.

Further topics include Newtonian correspondence, gravitational redshift and time dilation, stationary spacetimes, cosmological models of Friedmann-Lemaître-Robertson-Walker type, the expansion of the Universe, dark matter and dark energy, and the Schwarzschild and Kruskal solutions.

Addressed primarily to graduate students, researchers, and advanced readers in mathematics, physics, and mathematical physics, this book offers a compact yet mathematically precise pathway from tensor analysis on manifolds to the modern geometric understanding of gravitation. It is suitable both as a course text and as a reference for readers wishing to master the geometric methods that underlie contemporary General Relativity.