We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in terms of geometric realizations.

In [5], M. Tahara and Y. Watanabe constructed a family of almost Hermitian structures (J,G) on the tangent bundle TM of a Riemannian manifold and constructed a family of Hermitian and Kähler structure on the tangent bundle on a space form. It is well-known that there are sixteen classes of almost Hermitian manifolds ([3]). In this paper, we give the conditions for (J,G) such that TM belongs to each of these sixteen classes.

We give a definition of generalized Mannheim curve in Euclidean 4-space E⁴. We show some characterizations and examples of generalized Mannheim curves.

We will characterize the boundedness and compactness of weighted composition operators on the logarithmic Bloch spaces with iterated weights on the open unit disk.