The missing direction and differential geometry on Heisenberg manifolds

The purpose of this thesis is to investigate the conjugate points along the sub-Riemannian geodesics and to compute the Carnot-Caratheodory distance in the case of a step 4 sub-laplacian $DX=12X21+X22$ where $X1=6x1+4x2x26t$, $X2=6x2-4x1x26t$. The conjugate points are generating the t-axis and there are an infinite sub-Riemannian geodesics of different lengths which join the origin and any fixed point on the t-axis.; In the second part of the thesis we construct a Riemannian metric which extends the sub-Riemannian metric on the 3-dimensional Heisenberg manifolds on $R3$ and prove that the manifold is Einstein along the horizontal distribution, i.e. the Ricci tensor is proportional with the metric on the horizontal vector fields. Finally we estimate the occurrence of the first conjugate points on these geodesics on the Heisenberg group and on the 3-dimensional Heisenberg manifolds in general.