Syllabus for a courses on Differential Geometry of 42 hours given at JYU.
21 Lessons:
1. [canceled]
2. ? manifolds, tangent space, smooth maps between manifolds.
3. [no exercise session]
4.? Parameterized surfaces; regular surfaces; ; Examples: Taurus and Sphere; surfaces as a place of zeros
5.? tangent basis induced by local parameterization; the first basic form; metric coefficients; Differential of a smooth map, Isometries,
6.? Curves: Length of a curve. logarithmic spiral length, Parameterization with respect to the arc length. Existence of reparameterization with respect to the arc length.
1. Orientability of vector spaces and manifolds, Gauss maps for codimension-one sub manifolds, Gauss maps induced by local parametrizations, Symmetry of the differential of the Gauss map; Second fundamental form;
2. Curves: velocity of a curve, tangent versor. Curvature and normal unit vector of a PRLA curve. Formula for the curvature and normal unit vector for a regular curve.
3. Normal section;
4. Oriented curvature, Normal curvature in one direction, normal curvature of a curve, Meusnier' proposition, principal curvatures and principal directions; Euler's formula for the second fundamental form; Gaussian curvature and mean curvature; form coefficients
5. Gaussian curvature; symbols of Christoffel; Expression of Christoffel's symbols in terms of metric coefficients and their derivatives; Gauss's egregium theorem
6. Rotation surfaces; Tractor and pseudosphere
1. Geodesics; vector field along a curve; covariant derivative; parallelism; existence of parallel transport; existence and equation of geodesics; exponential map
2. properties of the exponential map; injection radius; geodesic balls and circumferences; radial geodesics; radial field and its properties;
3. Matrix groups and Lie groups; GL (n), Verify that SL (n) and O (n) are manifolds and calculate their size;
4. Gauss' lemma; geodesics as curves (locally) shorter.
5. Area and integral on surfaces; a relationship between Gaussian curvature and area growth;
6. minimum and minimizing surfaces, ruled surfaces.
1. Abstract differentiable manifolds, Terminology (local parameterization, local maps, local coordinates,…), derivations and tangent vectors, vector fields and Lie brackets
2. Riemannian metrics, covariant derivative and Levi-Civita connection.
3. Riemannian Geodesic equation