Search Results (27)

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized $\delta$-Casorati curvatures ${\delta }_c(t;q_1+q_2-1)$ and ${\widehat{\delta }}_c(t;q_1+q_2-1)$ for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples. Full article

In this paper, we utilize advanced optimization techniques on Riemannian submanifolds to establish two distinct inequalities concerning the generalized normalized $\delta$-Casorati curvatures of warped product pointwise semi-slant (WPPSS) submanifolds within complex space forms. We further identify the precise conditions under which these inequalities attain equality, providing valuable insights into their geometric and structural significance. Additionally, we also present results involving harmonic and Hessian functions, revealing a broader connection between curvature properties and analytic functions. Full article

We establish an improved Chen inequality involving scalar curvature and mean curvature and geometric inequalities for Casorati curvatures, on slant submanifolds in a Lorentzian–Sasakian space form endowed with a semi-symmetric non-metric connection. Also, we present examples of slant submanifolds in a Lorentzian–Sasakian space form. Full article

In this study, we focused on assessing the symmetry of shapes and quantifying an index of ‘order’ in three-dimensional shapes using curvature, which is important in product design. Specifically, the target three-dimensional shape was divided into two segments, and the Jensen–Shannon distance was calculated for the distribution of the Casorati curvatures in both segments to determine the similarity between them. This was proposed as an indicator of the ‘order’ exhibited by the shape. To validate the effectiveness of the proposed index, sensory evaluation experiments were conducted on three shapes: extruded, rotated, and vase. For the rotated shape, the coefficient of determination between the proposed index and the sensory evaluation value of ‘order’ on a 5-point Likert scale was found to be less than 0.1. The reason for the poor correlation coefficient of determination may be attributed to the bias in human perception, where individuals tend to perceive mirror symmetry with respect to the plane that includes the vertical axis when recognizing the mirror symmetry of an object. In contrast, for the extruded and vase shapes, the coefficients of determination were 0.36 and 0.66, respectively, supporting the validity of the proposed index. Nonetheless, the coefficient of determination decreased slightly for familiar extruded shapes and asymmetric vase shapes. In future research, our aim is to quantify ‘aesthetic preference’ by combining the ‘order’ and ‘complexity’ indexes. Full article

In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or $(\alpha ,\beta )$-type almost contact manifolds endowed with the Schouten–Van Kampen connection ($SVK$-connection), including generalized normalized $\delta$-Casorati Curvatures ($\delta$-CC). We also discuss submanifolds on which the equality situations occur. Lastly, we provided an example derived from this research. Full article

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic $II$-flat, isotropic minimal and isotropic $II$-minimal, the constant second Gaussian curvature, and the constant mean curvature of surfaces with constant negative curvature (SCNC) in the simply isotropic 3-space. Surfaces with symmetry are obtained when the mean curvatures are equal. Further, we have investigated the constant Casorati, the tangential and the amalgamatic curvatures of SCNC. Full article

Generative design is a system that automates part of the design process, but it cannot evaluate psychological issues related to shapes, such as “beauty” and “liking”. Designers therefore evaluate and choose the generated shapes based on their experience. Among the design features, “complexity” is considered to influence “aesthetic preference”. Although feature descriptors calculated from curvature can be used to quantify “complexity”, the selection guidelines for curvature and feature descriptors have not been adequately discussed. Therefore, this study aimed to conduct a systematic classification of curvature and a feature descriptor of 3D shapes and to apply the results to the “complexity” quantification. First, we surveyed the literature on curvature and feature descriptors and conducted a systematic classification. To quantify “complexity”, we used five curvatures (Gaussian curvature, mean curvature, Casorati curvature, shape index, and curvature index) and a feature descriptor (entropy of occurrence probability) obtained from the classification and compared them with the sensory evaluation values of “complexity”. The results showed that the determination coefficient between the quantified and sensory evaluation values of “complexity” was highest when the mean curvature was used. In addition, the Casorati curvature tended to show the highest signal-to-noise ratio (i.e., a high determination coefficient irrespective of the parameters set in the entropy calculation). These results will foster the development of generative design of 3D shapes using psychological evaluation. Full article

We propose fundamental inequalities for contact pseudo-slant submanifolds of $(ϵ)$-para Sasakian space form employing generalized normalized $\delta$-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants. Full article

In this paper, we establish some inequalities between the normalized $\delta$-Casorati curvatures and the scalar curvature (i.e., between extrinsic and intrinsic invariants) of spacelike statistical submanifolds in Sasaki-like statistical manifolds, endowed with a semi-symmetric metric connection. Moreover, we study the submanifolds satisfying the equality cases of these inequalities. We also present an appropriate example. Full article

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized $\delta$-Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant $\varphi$-sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, we investigate the equality cases of these inequalities. We also describe an illustrative example. Full article

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for $\delta (2,2)$-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end. Full article

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized $\delta$-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same ambient space. We also discuss the equality cases of the inequalities. Further, we provide some immediate geometric applications of the results. Finally, we construct some examples of slant and Legendrian submanifolds, respectively. Full article

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of $\delta$-invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on $\delta$-invariants and their applications have been summarized in the first author’s book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the $\delta$-invariants done mostly after the publication of the book. Full article

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated. Full article