Khazhgali Kozhasov
Summary: This course provides a comprehensive introduction to differential geometry, combining the classical theory of curves and surfaces with essential elements of smooth and Riemannian manifolds. The course begins with the local and global geometry of curves and surfaces in Euclidean space, including the theory of space curves, Frenet frames, curvature and torsion, total curvature, geodesics, minimal surfaces, and Gaussian curvature, with particular emphasis on the distinction between intrinsic and extrinsic geometry.
Basic notions of differentiable manifolds are introduced as needed, including tangent vectors, tangent spaces, and vector fields. The course then develops Riemannian metrics on surfaces, length of curves, geodesic distance, isometries, and the Levi–Civita connection, together with parallel transport, the exponential map, and normal coordinates. The course concludes with an introduction to curvature, including the Riemannian curvature tensor and sectional curvature, and selected classical results such as the Bonnet–Myers theorem.