Geometry of Manifolds II: Homework

1. Every vector bundle on a contractible manifold is trivial. Thus a contractible manifold is parallelizable.
2. Any Lie group is parallelizable.
3. A closed orientable 3-manifold is parallelizable. ( project problem )
4. Any principal bundle with a global section is trivial.
5. Let P -> B be a principal bundle with group G and suppose a representation G -> GL(V) is given.
   Let E=P  ×_G  V   ->   B   be the associated vector bundle. Verify that this is a vector bundle with fiber V and cocycles g_αβ : U_αβ -> G -> GL(V) , where U_αβ is the intersection of U_α and U_β.
6. Verify the equivalence of various definitions of connections on a vector bundle (i.e. horizontal distribution, horizontal lifts, horizontal liftings of curves, parallel transport, covariant derivative, etc.)
7. Show that the exponential map: sl₂(R) -> SL₂(R)   is not onto.
8. Suppose f: N -> M is a map and p: E -> M is a vector bundle equipped with a connection H. Verify that the inverse image of H defines a connection on f^*E indeed.
9. Let E -> B be a vector bundle equipped with a flat connection. Investigate the cohomology of the complex given by the exterior covariant derivative. Try some non-trivial examples.
10. Show that sectional curvature determines the curvature tensor uniquely.
11. The two definitions of Exp for compact linear Lie groups coincide ...
12. For a riemannian metric g with the riemannian connetcion, \nabla(g)=0, where \nabla is the covariant differentiation extended to symmetric 2-tensors.
13. Verify that ( \nabla _X α)(X₁, ..., X_r)= X.(α(X₁, ..., X_r)), where \nabla is the riemannian connection on the riemannian manifold M , α is an r-form on M and X_i are vector fields on M such that \nabla X_j=0 .
14. Find a geometric description for the Levi-Civita connection, when a riemannian metric is given.
15. Find conditions on the Christoffel symbols, under which they determine the Levi-Civita connection of a riemannian metric.
16. Suppose \nabla is a riemannaian connection. Show that div(Y)=tr(X -> \nabla_X Y) for vector fields X and Y.
17. Let \nabla be a torsion-free connection on a tangent bundle TM which is also equipped with an almost-complex structure J: TM -> TM , J²= -I, and a riemannain metric g. Consider the parallel transport induced by \nabla. This parallel transport induces an R-linear isomorphisms on the fibers. Show that
    1. If this isomorphism is C-linear, then J is integrable and M a complex manifold.
    2. If this isomorphism is an isometry (preserving the inner products), then \nabla is the riemannian connection of g .
    3. If h is the hermitian metric associated to g and J, and the mentioned isomorphism is an isometry with respect to h, then M is a Kaehler manifold.