Amrınder Pal Singh, C. Atindogbe, Rakesh Kumar, V. Jain
We study null hypersurfaces of indefinite K"{a}hler manifolds and by taking the advantages of the almost complex structure $J$, we select a suitable rigging $zeta$, which we call the $J-$rigging, on the null hypersurface. This suitable rigging enables us to build an associated Hermitian metric $breve{g}$ on the ambient space and which is restricted into a non-degenerated metric $widetilde{g}$ on the normalized null hypersurface. We derive Gauss-Weingarten type formulae for null hypersurface $M$ of an indefinite K"{a}hler manifold $overline{M}$ with a fixed closed Killing $J-$rigging for $M$. Later, we establish some relations linking the curvatures, null sectional curvatures, Ricci curvatures, scalar curvatures etc. of the ambient manifold and normalized null hypersurface.
{"title":"Normalized Null hypersurfaces of Indefinite K\"{a}hler Manifolds","authors":"Amrınder Pal Singh, C. Atindogbe, Rakesh Kumar, V. Jain","doi":"10.36890/iejg.1148612","DOIUrl":"https://doi.org/10.36890/iejg.1148612","url":null,"abstract":"We study null hypersurfaces of indefinite K\"{a}hler manifolds and by taking the advantages of the almost complex structure $J$, we select a suitable rigging $zeta$, which we call the $J-$rigging, on the null hypersurface. This suitable rigging enables us to build an associated Hermitian metric $breve{g}$ on the ambient space and which is restricted into a non-degenerated metric $widetilde{g}$ on the normalized null hypersurface. We derive Gauss-Weingarten type formulae for null hypersurface $M$ of an indefinite K\"{a}hler manifold $overline{M}$ with a fixed closed Killing $J-$rigging for $M$. Later, we establish some relations linking the curvatures, null sectional curvatures, Ricci curvatures, scalar curvatures etc. of the ambient manifold and normalized null hypersurface.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45971409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On the total space of the cotangent bundle of a Riemannian manifold, we construct a semi-Riemannian metric $G$, with respect to which an almost complex structure $J$ introduced by Oproiu and Porocb{s}niuc is self-adjoint. The structure $(J,G)$ turnes out to be an almost complex structure with Norden metric (this notion is known in the literature from Norden's papers). The semi-Riemannian context is different from the Riemannian one, as it is pointed out by Duggal and Bejancu in their monograph. We study this structure and provide some necessary and sufficient conditions for it to be a K"ahler structure with Norden metric.
{"title":"An almost Complex Structure with Norden Metric on the Phase Space","authors":"C. Bejan, G. Nakova","doi":"10.36890/iejg.1278651","DOIUrl":"https://doi.org/10.36890/iejg.1278651","url":null,"abstract":"On the total space of the cotangent bundle of a Riemannian manifold, we construct a semi-Riemannian metric $G$, with respect to which an almost complex structure $J$ introduced by Oproiu and Porocb{s}niuc is self-adjoint. The structure $(J,G)$ turnes out to be an almost complex structure with Norden metric (this notion is known in the literature from Norden's papers). The semi-Riemannian context is different from the Riemannian one, as it is pointed out by Duggal and Bejancu in their monograph. We study this structure and provide some necessary and sufficient conditions for it to be a K\"ahler structure with Norden metric.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43361611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we use Hessian comparison and volume comparison theorems to investigate the Mckean-type estimate theorem for the first eigenvalue of p-Laplacian and (p,q)-Laplacian operators on Finsler manifolds.
{"title":"Mckean-type Estimates for the First Eigenvalue of the $p$-Laplacian and $(p,q)$-Laplacian Operators on Finsler Manifolds","authors":"Sakineh Haji̇aghasi̇, S. Azami","doi":"10.36890/iejg.1133383","DOIUrl":"https://doi.org/10.36890/iejg.1133383","url":null,"abstract":"In this paper, we use Hessian comparison and volume comparison theorems to investigate the Mckean-type estimate theorem for the first eigenvalue of p-Laplacian and (p,q)-Laplacian operators on Finsler manifolds.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48172945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Carlos Avi̇la, Matias Navarro, O. Palmas, D. Solis
A null manifold is a differentiable manifold M endowed with a degenerate metric tensor g. In this work we provide sufficient conditions for a null manifold to be isometrically immersed as a hypersurface into a simple connected semi-Riemannian manifold of constant sectional curvature c and index q
{"title":"On Isometric Immersions of Null Manifolds into Semi-Riemannian Space Forms of Arbitrary Index","authors":"Carlos Avi̇la, Matias Navarro, O. Palmas, D. Solis","doi":"10.36890/iejg.1274307","DOIUrl":"https://doi.org/10.36890/iejg.1274307","url":null,"abstract":"A null manifold is a differentiable manifold M endowed with a degenerate metric tensor g. In this work we provide sufficient conditions for a null manifold to be isometrically immersed as a hypersurface into a simple connected semi-Riemannian manifold of constant sectional curvature c and index q","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44330563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the alpha circle inversion by using alpha distance function instead of Euclidean distance in definition of classical inversion. We give some proporties of alpha circle inversion. Also this new transformation is applied to well known fractals. Then new fractal patterns are obtained. Moreover we generalize the method called circle inversion fractal be means of the alpha circle inversion. In alpha plane, we give a generalization of alpha circle inversion fractal by using the concept of star-shaped set inversion which is a generalization of circle inversion fractal.
{"title":"Inversions and Fractal Patterns in Alpha Plane","authors":"Ö. Gelişgen, T. Ermiş","doi":"10.36890/iejg.1244520","DOIUrl":"https://doi.org/10.36890/iejg.1244520","url":null,"abstract":"In this paper, we introduce the alpha circle inversion by using alpha distance function instead of Euclidean distance in definition of classical inversion. We give some proporties of alpha circle inversion. Also this new transformation is applied to well known fractals. Then new fractal patterns are obtained. Moreover we generalize the method called circle inversion fractal be means of the alpha circle inversion. In alpha plane, we give a generalization of alpha circle inversion fractal by using the concept of star-shaped set inversion which is a generalization of circle inversion fractal.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45190642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a non-trivial generalized $m$-quasi Einstein manifold $M$ with finite $m$ and associated divergence-free affine Killing vector field, and show that $M$ reduces to an $m$-quasi Einstein manifold. In addition, if $M$ is complete, then it splits as the product of a line and an $(n-1)$-dimensional negatively Einstein manifold. Finally, we show that the same result holds for a complete non-trivial $m$-quasi Einstein manifold $M$ with finite $m$ and associated affine Killing vector field.
{"title":"Rigidity Results On Generalized m-Quasi Einstein Manifolds with Associated Affine Killing Vector Field.","authors":"Rahul Poddar, B. Subramanian, R. Sharma","doi":"10.36890/iejg.1286128","DOIUrl":"https://doi.org/10.36890/iejg.1286128","url":null,"abstract":"We study a non-trivial generalized $m$-quasi Einstein manifold $M$ with finite $m$ and associated divergence-free affine Killing vector field, and show that $M$ reduces to an $m$-quasi Einstein manifold. In addition, if $M$ is complete, then it splits as the product of a line and an $(n-1)$-dimensional negatively Einstein manifold. Finally, we show that the same result holds for a complete non-trivial $m$-quasi Einstein manifold $M$ with finite $m$ and associated affine Killing vector field.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47571914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Spacelike and timelike isotropic submanifolds of pseudo-Riemannian spaces have interesting properties, with important applications in Mathematics and Physics. The article presents inequalities for isotropic spacelike and timelike submanifolds of pseudo-Riemannian space forms and isotropic Lorentzian submanifolds are also considered.
{"title":"Inequalities on Isotropic Submanifolds in Pseudo-Riemannian Space Forms","authors":"Alexandru Ciobanu","doi":"10.36890/iejg.1260464","DOIUrl":"https://doi.org/10.36890/iejg.1260464","url":null,"abstract":"Spacelike and timelike isotropic submanifolds of pseudo-Riemannian spaces have interesting properties, with important applications in Mathematics and Physics. The article presents inequalities for isotropic spacelike and timelike submanifolds of pseudo-Riemannian space forms and isotropic Lorentzian submanifolds are also considered.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47246381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish some properties of the $k$-slant and pointwise $k$-slant submanifolds of an almost contact metric manifold with a special view towards the integrability of the component distributions. We prove some results for totally geodesic pointwise $k$-slant submanifolds. Furthermore, we obtain some nonexistence results for pointwise $k$-slant submanifolds in the almost contact metric setting.
{"title":"On Pointwise $k$-slant Submanifolds of Almost Contact Metric Manifolds","authors":"A. Blaga, D. Laţcu","doi":"10.36890/iejg.1274538","DOIUrl":"https://doi.org/10.36890/iejg.1274538","url":null,"abstract":"We establish some properties of the $k$-slant and pointwise $k$-slant submanifolds of an almost contact metric manifold with a special view towards the integrability of the component distributions. We prove some results for totally geodesic pointwise $k$-slant submanifolds. Furthermore, we obtain some nonexistence results for pointwise $k$-slant submanifolds in the almost contact metric setting.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46291763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we discuss the stability of some anti-invariant submanifolds of golden Riemannian manifolds under certain conditions in terms of the Ricci curvature tensors of the ambient manifold and the submanifold.
{"title":"The Stability Problem of Certain Anti-Invariant Submanifolds in Golden Riemannian Manifolds","authors":"Mustafa Gök, E. Kiliç","doi":"10.36890/iejg.1240437","DOIUrl":"https://doi.org/10.36890/iejg.1240437","url":null,"abstract":"In this study, we discuss the stability of some anti-invariant submanifolds of golden Riemannian manifolds under certain conditions in terms of the Ricci curvature tensors of the ambient manifold and the submanifold.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47102385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is twofold. Firstly, we will investigate the link between the condition for the functions $phi(s)$ from $(alpha, beta)$-metrics of Douglas type to be self-concordant and k-self concordant, and the other objective of the paper will be to continue to investigate the recently new introduced $(alpha, beta)$-metric ([17]): $$ F(alpha,beta)=frac{beta^{2}}{alpha}+beta+a alpha $$ where $alpha=sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric; $beta=b_{i}y^{i}$ is a 1-form, and $ain left(frac{1}{4},+inftyright)$ is a real positive scalar. This kind of metric can be expressed as follows: $F(alpha,beta)=alphacdot phi(s)$, where $phi(s)=s^{2}+s+a$. In this paper we will study some important results in respect with the above mentioned $(alpha, beta)$-metric such as: the Kropina change for this metric, the Main Scalar for this metric and also we will analyze how the condition to be self-concordant and k-self-concordant for the function $phi(s)$, can be linked with the condition for the metric $F$ to be of Douglas type. self-concordant functions, Kropina change, main scalar.
{"title":"Some Aspects on a Special Type of $(alpha,beta )$-metric","authors":"Laurian-loan Piscoran, C. Barbu","doi":"10.36890/iejg.1265041","DOIUrl":"https://doi.org/10.36890/iejg.1265041","url":null,"abstract":"The aim of this paper is twofold. Firstly, we will investigate the link between the condition for the functions $phi(s)$ from $(alpha, beta)$-metrics of Douglas type to be self-concordant and k-self concordant, and the other objective of the paper will be to continue to investigate the recently new introduced $(alpha, beta)$-metric ([17]):\u0000 $$\u0000 F(alpha,beta)=frac{beta^{2}}{alpha}+beta+a alpha\u0000 $$\u0000 where $alpha=sqrt{a_{ij}y^{i}y^{j}}$ is a Riemannian metric; $beta=b_{i}y^{i}$ is a 1-form, and $ain left(frac{1}{4},+inftyright)$ is a real positive scalar. This kind of metric can be expressed as follows: $F(alpha,beta)=alphacdot phi(s)$, where $phi(s)=s^{2}+s+a$.\u0000 In this paper we will study some important results in respect with the above mentioned $(alpha, beta)$-metric such as: the Kropina change for this metric, the Main Scalar for this metric and also we will analyze how the condition to be self-concordant and k-self-concordant for the function $phi(s)$, can be linked with the condition for the metric $F$ to be of Douglas type.\u0000\u0000self-concordant functions, Kropina change, main scalar.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44185088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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