Geometry 4

COMPLEMENTS OF TOPOLOGY

- Metric spaces

- Compactness, countability axioms, paracompactness, separability


DIFFERENTIABLE MANIFOLDS

- Differentiable manifolds

- Maps between manifolds

- The tangent space

- Rank theorem, inverse function theorem and applications; immersions, submersions, embeddings and submanifolds

- The tangent and cotangent bundles; vector fields

- Vector bundles (some notes)

- Multilinear algebra and tensor fields

- Differential forms and exterior differential

- Orientable manifolds and integration on manifolds

- Riemannian manifolds (some notes)

CURVES
Curves in the plane
Vector product and curves in the space. Arc Length
Frenet Triedron. Rigidity theorem of curves in the plane and in the space.
SURFACES
Definitions of regular surface, tangent space. First Fundamental form, length, angles and area on a surface. Differential maps between surfaces, the differential map, the Gauss map, Weingarten operator, principal curvatures, Principal directions, Second Fundamental form, Gauss curvature. Nature of points on a surface. Normal curvature, Meusnier Theorem. Asymptotic curves, lines of curvature, Rigidity theorem of a surface. Isometries between surfaces. Egregium Theorem of Gauss. Intrinsic concepts. Covariant derivatives, geodetic curvatures. Ideas of the Gauss Bonnet Theorem and applications.