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Zusammenfassung:
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The only non-compact four-dimensional rank-one symmetric spaces are the complex
hyperbolic plane CH2 and the four-dimensional real hyperbolic space RH4. As
connected homogeneous manifolds of negative sectional curvature, these spaces
admit the structure of a four-dimensional real solvable Lie group equipped with
a left-invariant metric. This Lie group appears naturally in the Poincar´e half-space
model of real hyperbolic space and in the Siegel paraboloid model of the complex
hyperbolic plane. The boundary of the paraboloid model carries the structure of the
Heisenberg group.
Hermitian structures consist of a left-invariant Riemannian metric together with
a compatible complex structure. In this thesis, all such structures are classified
and their geometric properties are studied. It is shown that every Riemannian
metric on real hyperbolic space admits a two-dimensional sphere of Hermitian
complex structures. In the case of the complex hyperbolic plane, some metrics
admit exactly four distinct Hermitian complex structures, while others admit a
two-dimensional sphere of such structures. Their curvature properties, holonomy
groups, and self-duality are investigated. It is shown that the standard metric on
the complex hyperbolic plane is the unique K¨ahler metric within the obtained
classification, whereas all Riemannian metrics on real hyperbolic space are Einstein.
Geodesics on the solvable Lie groups of the spaces CH2 and RH4, with respect
to all possible left-invariant Riemannian metrics, are studied in this thesis using
the Euler–Arnold equations. These equations effectively reduce a system of secondorder
differential equations on a Lie group to a system of first-order equations on
the corresponding Lie algebra. Numerical solutions of these equations enable the
visualization of geodesics and geodesic spheres. |