Topological manifolds and topological manifolds with boundary. Topological surfaces. Differentiable manifolds; examples (n-spheres, tori, projective spaces). Differentiable mappings. Rank theorem. Submanifolds. Local theory of curves in R3. Curvature and torsion. Frenet Serret formulas. Local theory of surfaces in R3. First fondamental form. Examples. Arc lengh of a curve on a surface. Gauss map. Second fondamental form. Normal curvature of a curve on a surface. Principal curvatures. Gaussian curvature, average curvature. Elliptic, parabolic and hyperbolic points. Tangent spaces. Differential map.
Geometria 4 mod/02
Partition of Unity. Vector bundles and sections. Tangent and cotangent bundles. Differential forms. Exterior differentiation. Volume form and orientation. Integration on manifolds. Stokes' theorem.