Bayram Şahin

In this work, we investigate curvature properties of Sasakian Hom-Lie algebras. Using the Hom-Levi- Civita connection on metric Hom-Lie algebras, we obtain explicit expressions for the curvature and Ricci tensors associated with a Sasakian Hom-Lie structure. We derive the necessary and sufficient differential condition for the Ricci tensor to be 𝜂-parallel. Furthermore, we establish fundamental curvature identities and provide a characterization of the 𝜂-Einstein condition. These results provide a natural extension of the curvature theory of Sasakian Lie algebras to the framework of twisted algebraic structures.

Recently, the PTR-submanifold of an almost Hermitian manifold has been defined and studied as a large class including CR-submanifolds, hemi-slant submanifolds, and pointwise hemi-slant submanifolds, which are well-known in the literature. This novel category of submanifolds comprises two distributions, one of which is antiinvariant while the complementary distribution is arbitrary. This paper discusses the case of a PTR-submanifold being a warped product. It is first shown that there are no such warped product submanifolds for the case where the first factor is antiinvariant. Therefore, warped product PTR-submanifolds whose second factor is antiinvariant are examined. A characterization is given for this second case, and an inequality is obtained in terms of the second fundamental form and warping function. Additionally, the situation in which the ambient space of the warped product PTR-submanifold is a complex space form is also studied in detail. In this direction, a new version of the inequality is found, and the corresponding geometric results are given accordingly.

Distance metric selection remains a fundamental challenge in machine learning, as no single metric universally captures the geometric structure of all datasets. In this study, we propose a novel supervised metric learning framework, termed Eigenvalue-Weighted Distance Fusion (EWDF), that constructs a dataset-specific composite metric through principal component analysis of intra-class pairwise distance representations. For each dataset, all same-class sample pairs are used to build a new feature space in which each dimension corresponds to a distance function, namely Euclidean, Manhattan, Minkowski (p=3), Chebyshev, and Canberra. Principal component analysis is then applied to this pairwise distance feature space, and the resulting eigenvalues are used as importance weights. Only distance functions associated with positive eigenvalue contributions are retained, and their positively weighted linear combination yields a valid, dataset-adaptive composite metric. The mathematical validity of the proposed metric is guaranteed by the non-negativity of the eigenvalue-derived coefficients and the metric properties of the constituent distance functions. Computational experiments are conducted on 13 benchmark datasets spanning three different data characteristics: balanced, imbalanced, and outlier-contaminated distributions. The proposed method is benchmarked against four established metric learning approaches. Experimental results demonstrate that EWDF achieves superior classification performance in the majority of cases across all dataset categories, with particularly notable improvements observed on imbalanced and outlier-contaminated datasets. These findings suggest that eigenvalue-weighted distance fusion provides a principled, interpretable, and computationally efficient alternative to existing metric learning paradigms, with broad applicability to distance-sensitive learning tasks.

In this study we examine the cases of constant sectionalcurvature in Metallic Riemannian manifolds. We show that classical andholomorphic-like curvatures generally make the manifold at. Based on this, we introduce a new concept called Metallic sectional curvature,and in this setting, we derive the explicit expression of the curvaturetensor. In addition, the Ricci tensor and scalar curvature are computed,and the conditions under which the manifold is Einstein are discussed.Finally, this new curvature notion is applied to semi-invariant and totallyumbilical submanifolds, yielding rich geometric results.

In this paper, we introduce Riemannian submersions of a hemi-slant submanifold of a Kähler manifold by observing the integrability of the anti-invariant distribution of a hemi-slant submanifold and the integrability of the vertical distribution of a Riemannian submersion. Using this notion, we show that the base manifold is a Kähler manifold in the submersion of a hemi-Kaehlerian slant submanifold of an almost Hermitian manifold. We obtain an inequality between the sectional curvature of the hemi-Kaehlerian slant submanifold and the holomorphic sectional curvature of the base manifold. If this inequality becomes equality, a geometric result is given. In addition, the Ricci tensor field on the horizontal distribution along this submersion is also found.

In this paper, Riemannian maps from a CR-submanifold of an almost Hermitian manifold to another almost Hermitian manifold are studied. Such Riemannian maps include the special class of CR-submersions, which are well known in the literature. First, the notion of Riemannian map from CR-submanifold to an almost Hermitian manifold is presented and the invariant of the image spaces of these maps is shown. The harmonicity of these maps is investigated. Also, the relation between the holomorphic sectional curvature of the target manifold and the holomorphic sectional curvature of the CR-submanifold is obtained depending on the second fundamental form of the submanifold and the second fundamental form of the Riemannian map. Finally, the Ricci curvature of the ambient manifold of the CR-submanifold in the direction of the invariant distribution is established in terms of the Ricci curvature of the image space, the Ricci curvature of the anti-invariant distribution, the mean curvature vector fields of the holomorphic distribution and the anti-invariant distribution.

In this talk, the geometry of a surface is examined when the shape operator matrix of a surface along a surface curve is almost complex structure, almost product structure and almost tangent structure. First, it was shown that the shape operator matrix of a surface along a surface curve cannot be almost complex structure. Then, two cases were considered when the shape operator matrix of a surface along a surface curve is almost product structure. First, when the geodesic curvature of the surface is different from zero, the Gaussian curvature, mean curvature, umbilicality of the surface were examined and the principal curvatures were found. In the second case, when the geodesic curvature of the surface is zero, the character of the surface was examined. Finally, the Gaussian curvature, mean curvature and principal curvatures of the surface are determined in the case where the shape operator matrix is almost tangent structure.

In this paper, we introduce a new submanifold of a Kähler manifold. This new class of submanifolds includes totally real submanifolds, proper CR submanifolds, hemi-slant submanifolds and pointwise hemi-slant submanifolds as special classes. After giving examples for this new class of submanifolds, the geometry of distributions and their maximal integral manifolds is investigated. Then, the effect of morphisms defined by distributions on the geometry of the submanifold is studied. In the case that the ambient space in which the submanifold is contained is a complex space form, we determine special cases where both the submanifold and the ambient space are reduced. Furthermore, we determine the position of the mean curvature vector field when the submanifold is totally umbilical, and with the help of this, the geometric structure of the submanifold is obtained. Finally, if this new class of submanifolds is a Riemannian product manifold, the non-existence of this product submanifold in complex hyperbolic space is shown and an inequality related to the second fundamental form for complex projective space is given. It is shown that the factors of the Riemannian product manifold are also totally geodesic in complex projective space if the inequality becomes the equality.