Ricci Curvatures on Hypersurfaces of Almost Product-like Statistical Manifolds

Yıl 2024, Cilt: 9 Sayı: 1, 33 - 46, 30.04.2024

Mehmet Gülbahar , Esra Erkan , Meral Düzgör

https://doi.org/10.30931/jetas.1419021

Öz

Riemannian curvature invariants on hypersurfaces of an almost product-like manifold with constant curvature υ are computed. Various relationships involving sectional curvatures and Ricci curvatures have been obtained. Using the Chen-Ricci inequality, some characterizations are presented.

Anahtar Kelimeler

Riemannian curvature invariants, hypersurfaces, product manifolds, Chen-Ricci inequality

Etik Beyan

Bu çalışmanın, özgün bir çalışma olduğunu; çalışmanın hazırlık, veri toplama, analiz ve bilgilerin sunumu olmak üzere tüm aşamalarından bilimsel etik ilke ve kurallarına uygun davrandığımı; bu çalışma kapsamında elde edilmeyen tüm veri ve bilgiler için kaynak gösterdiğimi ve bu kaynaklara kaynakçada yer verdiğim kabul ederek etik görev ve sorumluluklara riayet ettiğimi beyan ederim.

Destekleyen Kurum

Scientific and Technological Research Council of Turkey (TÜBİTAK)

Proje Numarası

122F326

Teşekkür

This study is supported by the Scientific and Technological Research Council of Turkey (TÜBiTAK)

Kaynakça

• [1] Chen, B.-Y., "Strings of Riemannian invariants, inequalities, ideal immersions and their applications", In: The Third Pacific Rim Geometry Conference (Seoul 1996), Int. Press, Cambridge MA, 25 (1998) : 7–60.
• [2] Chen, B.-Y., "Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions", Glasgow Mathematical Journal 41 (1999) : 33–41.
• [3] Chen, B.-Y., "On Ricci curvature of isotropic and Langrangian submanifolds in complex space forms", Archiv der Mathematik (Basel) 74 (2000) : 154–160.
• [4] Chen, B.-Y, "Pseudo-Riemannian geometry, δ-invariants and applications", World Scientific (2011).
• [5] Suceava B. D., Vajiac M. B., "Remarks on Chen's fundamental inequality with classical curvature invariants in Riemannian spaces", Analele Stiintifice ale Universitatii Alexandru Ioan Cuza" Din Iasi. Mathematics. (NS) 54 (2008) : 27-37.
• [6] Chen B.-Y, Blaga A. M., Vîlcu G-E., "Differential geometry of submanifolds in complex space forms involving δ-invariants", Mathematics 10 (2022): 591.
• [7] Akyol M. A., Demir R., Poyraz N. Ö., Vîlcu, G-E., "Optimal inequalities for hemi-slant Riemannian submersions", Mathematics, 10 (2022) : 3993.
• [8] Özgür C., and Tripathi M. M., "On submanifolds satisfying Chen's equality in a real space form", The Arabian Journal for Science and Engineering 33(2A) (2008) : 321-330.
• [9] Gülbahar M., Kılıç, E. Keleş, S., "A useful orthonormal basis on bi-slant submanifolds of almost Hermitian manifolds", Tamkang Journal of Mathematics 47(2) (2016) : 143-161.
• [10] Sarı R., Ünal İ. "On curvatures of semi-invariant submanifolds of Lorentzian para-Sasakian manifolds", Turkish Journal of Mathematics and Computer Science 15(2) (2023) : 464-469.
• [11] Amari, S. "Information geometry of the EM and em algorithms for neural networks", Neural Networks 8(9) (1995): 1379–1408.
• [12] Balan V., Grushevskaya H.V., Krylova N.G., Krylov G.G., Lipnevich I., "Two dimensional first-order phase transition as signature change event in contact statistical manifolds with Finsler metric", Applied Sciences 21 (2019): 11–26.
• [13] Caticha A., "Geometry from information geometry", AIP Conference Proceedings, AIP Publishing, 1757(1) (2016).
• [14] Sun K., Marchand-Maillet S., "An information geometry of statistical manifold learning", Proceedings of the 31st International Conference on Machine Learning (ICML-2014), (2014) : 1-9.
• [15] Rao C. R., "Information and accuracy attainable in the estimation of statistical parameters", Bulletin of the Calcutta Mathematical Society 37 (1945), 81-91.
• [16] Aytimur H., Kon M., Mihai A., Özgür, C., Takano, K., "Chen inequalities for statistical submanifolds of Kaehler-like statistical manifolds", Mathematics 7(12) (2019): 1202.
• [17] Bahadır O., Siddiqui A. N., Gülbahar M., Alkhaldi, A. H., "Main curvatures identities on lightlike hypersurfaces of statistical manifolds and their characterizations", Mathematics 10(13) (2022) : 2290.
• [18] Erkan, E., Gülbahar, M., "Chen's basic inequalities for hypersurfaces of statistical Riemannian manifolds", International Journal of Maps in Mathematics 6(1) (2023) : 37-53.
• [19] Erkan, E., Gülbahar, M., "Locally product-like statistical manifolds and their hypersurfaces", International Electronic Journal of Geometry, 16(2) (2023) : 435-450.
• [20] Feng W. U., Jiang Y., Zhang L., "Some results on statistical hypersurfaces of Sasakian statistical manifolds and holomorphic statistical manifolds", International Electronic Journal of Geometry 14(1) (2021): 46-58.
• [21] Furuhata H., "Hypersurfaces in statistical manifolds", Differential Geometry and its Applications 27(3) (2009): 420-429.
• [22] Furuhata H., "Statistical hypersurfaces in the space of Hessian curvature zero", Differential Geometry and its Applications 29(1) (2011): S86–S90.
• [23] Siddiqui A. N., Shahid M. H., "Optimizations on statistical hypersurfaces with Casorati curvatures", Kragujevac Journal of Mathematics 45(3) (2021): 449-463.
• [24] Kazan S. and Takano K. "Anti-invariant holomorphic statistical submersions", Results in Mathematics 78(128) (2023) : 1-18.
• [25] Adati T., "Submanifolds of an almost product Riemannian manifold", Kodai Mathematical Journal 4(2) (1981): 327-343.
• [26] Atçeken M., Şahin B., Kılıç E., "On invariant submanifolds of Riemannian warped product manifold", Turkish Journal of Mathematics 27(3) (2003): 407-423.
• [27] Gündüzalp Y., "Slant submersions from almost product Riemannian manifolds", Turkish Journal of Mathematics 37(5) (2013): 863-873.
• [28] Gündüzalp Y., "Semi-slant submersions from almost product Riemannian manifolds", Demonstratio Mathematica 49(3) (2016) : 345-356.
• [29] Gündüzalp Y., "Anti-invariant Riemannian submersions from almost product Riemannian manifolds", Mathematical Sciences and Applications E-Notes 1(1) (2013) : 58-66.
• [30] Li Y., Alkhaldi A. H., Ali A., "Geometric mechanics on warped product semi-slant submanifold of generalized complex space forms", Advances in Mathematical Physics (2021): 1-15.
• [31] Li Y., Srivastava S. K., Mofarreh F., Kumar A., Ali A., "Ricci Soliton of CR-Warped Product Manifolds and Their Classifications", Symmetry 15(5) (2023): 976.
• [32] Rahman S., Haseeb A., Almusawa H., "Geometry of warped product pseudo slant submanifolds in a nearly quasi-Sasakian manifold", Italian Journal of Pure and Applied Mathematics, (48) (2022) : 1021–1032.
• [33] Şahin B., Atçeken M., "Semi-invariant submanifolds of Riemannian product manifold", Balkan Journal of Geometry and its applications 8(1) (2023) : 91-100.
• [34] Şahin B., "Slant submanifolds of an almost product Riemannian manifold", Journal of the Korean Mathematical Society 43(4) (2006) : 717-732.
• [35] Opozda B., "A sectional curvature for statistical structures", Linear Algebra and its Applications 497, (2016) : 134–161.