Differential Geometry

Basic definitions, examples, coordinate charts, atlases, and differentiable structures.

Smooth maps, tangent vectors, derivatives of smooth maps.

Tangent vectors, cotangent vectors, one-forms, vector bundles.

Tangent/cotangent bundles; partitions of unity and their uses.

Submersions, immersions, embeddings, submanifolds, Sard’s theorem.

Inverse/implicit function theorems, rank theorem, Whitney embedding, regular values.

Vector fields, Lie brackets, flows, Lie groups as manifolds, exponential map.

Integral curves, flows, pushforward of vector fields.

Exterior algebra, differential forms, wedge product, Stokes’ theorem.

Tensors, exterior forms, wedge product, pullbacks of forms.

Exterior differentiation, Lie derivatives, a first look at de Rham cohomology.

Cohomology, Poincaré duality, Mayer–Vietoris, orientations, integration on manifolds.

Cobordism, boundary operator, degree of a smooth map.

Riemannian metrics as smoothly varying inner products; examples and submanifolds.

Affine connections; Levi–Civita as metric-compatible and torsion-free; comparing vectors in different tangent spaces.

Geodesics as zero-acceleration curves; exponential map; construction of normal coordinates.

Riemann curvature tensor, sectional curvature and its geometry; symmetry identities.

Jacobi fields, second variation formula, conjugate points and stability of geodesic flows.

Equivalence of metric completeness, geodesic completeness, and compactness of closed balls.

Models with constant sectional curvature and their isometry groups; global behaviour.

Implications of nonpositive curvature; universal covers and diffeomorphism to $\mathbb{R}^n$.

Consequences of positive Ricci curvature: compactness and diameter estimates.

Relations between curvature, orientability, and the fundamental group in even dimensions.