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We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanv{z}urova are genuine generalizations of the ordinary notion of $k$-curvature homogeneity. The homothety group plays an essential role in the analysis.
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We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanzurova are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis. We give a complete classification of homothety homogeneous manifolds which are not homogeneous and which are not VSI and show that such manifolds are cohomogeneity one. We also give a complete description of the local geometry if the homothety character defines a split extension.
k-Curvature homogeneous three-dimensional Walker metrics are described for k=0,1,2. This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature. Moreover, potential functions are constructed in all the locally homogeneous manifolds resulting in steady gradient Ricci and Cotton solitons.
The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity three in dimension four. We show that these conditions are compatible with non-negative sectional curvature only if either the manifold is diffeomorphic to $mathbb{R}^n$ or the universal cover is an isometric product with a Euclidean factor. Moreover, we show that finite volume manifolds with conullity 3 are locally products.
We start with introducing one of the most fundamental notions of differential geometry, Manifolds. We present some properties and constructions such as submanifolds, tangent spaces and the tangent map. Then we continue with introducing the real and complex projective space, and describe them from some different points of view. This part is finished by showing that CP^n is a Grassmannian manifold. At this stage we are ready to present the main subject of this thesis. The Schwarzian curvature, usually seems to be an accidental by-product of the calculations, can be seen as a geometric interpretation of the Schwarzian derivative. Flanders interpreted the Schwarzian derivative of a C function as a curvature for curves in the projective line by using the moving frame method of Cartan. The same argumentation was extended by Gao to obtain the Schwarzian curvatures for curves in higher dimensional projective spaces. I give detailed presentation of Gaos work, where he presented the general formulas for the Schwarzian curvatures for curves in CP^n and gives some properties for the behaviour of the formulas, for example the transformation rules under change of coordinates. The Schwarzian curvatures for curves in CP, CP^2 and CP^3 are calculated, and some examples are given.
We study harmonic maps from a 3-manifold with boundary to $mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $pi / 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.
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