Pub Date : 2021-06-24DOI: 10.1515/advgeom-2021-0015
Theoharis Theofanidis
Abstract We aim to classify the real hypersurfaces M in a Kaehler complex space form Mn(c) satisfying the two conditions φl=lφ, $varphi l=lvarphi ,$where l=R(⋅,ξ)ξ and φ $l=R(cdot ,xi )xi text{ and }varphi $is the almost contact metric structure of M, and (∇ξl)X= $left( {{nabla }_{xi }}l right)X=$ω(X)ξ, where where ω(X) is a 1-form and X is a vector field on M. These two conditions imply that M is a Hopf hypersurface and ω = 0.
{"title":"Real hypersurfaces of non-flat complex space forms with two generalized conditions on the Jacobi structure operator","authors":"Theoharis Theofanidis","doi":"10.1515/advgeom-2021-0015","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0015","url":null,"abstract":"Abstract We aim to classify the real hypersurfaces M in a Kaehler complex space form Mn(c) satisfying the two conditions φl=lφ, $varphi l=lvarphi ,$where l=R(⋅,ξ)ξ and φ $l=R(cdot ,xi )xi text{ and }varphi $is the almost contact metric structure of M, and (∇ξl)X= $left( {{nabla }_{xi }}l right)X=$ω(X)ξ, where where ω(X) is a 1-form and X is a vector field on M. These two conditions imply that M is a Hopf hypersurface and ω = 0.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46450450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-24DOI: 10.1515/advgeom-2021-0017
Saeed Tafazolian, F. Torres
Abstract Let F be the finite field of order q2. In this paper we continue the study in [24], [23], [22] of F-maximal curves defined by equations of type yn=xℓ(xm+1). ${{y}^{n}}={{x}^{ell }}left( {{x}^{m}}+1 right).$New results are obtained via certain subcovers of the nonsingular model of vN=ut2−u ${{v}^{N}}={{u}^{{{t}^{2}}}}-u$where q = tα, α ≥ 3 is odd and N = (tα + 1)/(t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve.
{"title":"The curve Yn = Xℓ(Xm + 1) over finite fields II","authors":"Saeed Tafazolian, F. Torres","doi":"10.1515/advgeom-2021-0017","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0017","url":null,"abstract":"Abstract Let F be the finite field of order q2. In this paper we continue the study in [24], [23], [22] of F-maximal curves defined by equations of type yn=xℓ(xm+1). ${{y}^{n}}={{x}^{ell }}left( {{x}^{m}}+1 right).$New results are obtained via certain subcovers of the nonsingular model of vN=ut2−u ${{v}^{N}}={{u}^{{{t}^{2}}}}-u$where q = tα, α ≥ 3 is odd and N = (tα + 1)/(t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41640304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-17DOI: 10.1515/advgeom-2021-0023
Ming Xu
Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.
{"title":"Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition","authors":"Ming Xu","doi":"10.1515/advgeom-2021-0023","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0023","url":null,"abstract":"Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45435115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-21DOI: 10.1515/advgeom-2022-0011
L. Fukshansky, Siki Wang
Abstract Let L be a full-rank lattice in ℝd and write L+ for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L+. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L+, and we investigate the distribution of the gaps of S(X) in L+, i.e. the points in L+ ∖ S(X). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of L+ and of L+ ∖ S(X), for which we produce bounds in the spirit of Minkowski’s successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil heights of totally positive sub-semigroups of ideals in totally real number fields.
{"title":"Positive semigroups in lattices and totally real number fields","authors":"L. Fukshansky, Siki Wang","doi":"10.1515/advgeom-2022-0011","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0011","url":null,"abstract":"Abstract Let L be a full-rank lattice in ℝd and write L+ for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L+. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L+, and we investigate the distribution of the gaps of S(X) in L+, i.e. the points in L+ ∖ S(X). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of L+ and of L+ ∖ S(X), for which we produce bounds in the spirit of Minkowski’s successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil heights of totally positive sub-semigroups of ideals in totally real number fields.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42147775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-18DOI: 10.1515/advgeom-2022-0012
Paula Escorcielo, Daniel Perrucci
Abstract We extend previous results about Putinar’s Positivstellensatz for cylinders of type S × ℝ to sets of type S × ℝr in some special cases, taking into account r and the degree of the polynomial with respect to the variables moving in ℝr (this is to say, in the non-bounded directions). These special cases are in correspondence with the ones where the equality between the cone of non-negative polynomials and the cone of sums of squares holds. Degree bounds are provided.
{"title":"A few more extensions of Putinar’s Positivstellensatz to non-compact sets","authors":"Paula Escorcielo, Daniel Perrucci","doi":"10.1515/advgeom-2022-0012","DOIUrl":"https://doi.org/10.1515/advgeom-2022-0012","url":null,"abstract":"Abstract We extend previous results about Putinar’s Positivstellensatz for cylinders of type S × ℝ to sets of type S × ℝr in some special cases, taking into account r and the degree of the polynomial with respect to the variables moving in ℝr (this is to say, in the non-bounded directions). These special cases are in correspondence with the ones where the equality between the cone of non-negative polynomials and the cone of sums of squares holds. Degree bounds are provided.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41933262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-10DOI: 10.1515/advgeom-2021-0038
P. Caldiroli
Abstract We construct families of smooth functions H : ℝn+1 → ℝ such that the Euclidean (n + 1)-space is completely filled by not necessarily round hyperspheres of mean curvature H at every point.
{"title":"An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space","authors":"P. Caldiroli","doi":"10.1515/advgeom-2021-0038","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0038","url":null,"abstract":"Abstract We construct families of smooth functions H : ℝn+1 → ℝ such that the Euclidean (n + 1)-space is completely filled by not necessarily round hyperspheres of mean curvature H at every point.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45196956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-06DOI: 10.1515/advgeom-2021-0029
Sean Eberhard
Abstract Here is a simplified proof that every sharply transitive subset of PGL2(K) is a coset of a subgroup, for every finite field K.
摘要本文给出了PGL2（K）的每一个锐传递子集都是子群的陪集的简化证明。
{"title":"Sharply transitive sets in PGL2(K)","authors":"Sean Eberhard","doi":"10.1515/advgeom-2021-0029","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0029","url":null,"abstract":"Abstract Here is a simplified proof that every sharply transitive subset of PGL2(K) is a coset of a subgroup, for every finite field K.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46882854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-11DOI: 10.1515/advgeom-2020-0032
Zaili Yan
Abstract We develop a variational method to find pseudo-algebraic Ricci solitons on connected Lie groups.As applications, we prove that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and that any algebraic Ricci soliton on a semi-simple Lie group is Einstein. Furthermore, we construct several Lorentz algebraic Ricci solitons on the nilpotent Lie groups which have a codimension one abelian ideal.
{"title":"Pseudo-algebraic Ricci solitons on Einstein nilradicals","authors":"Zaili Yan","doi":"10.1515/advgeom-2020-0032","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0032","url":null,"abstract":"Abstract We develop a variational method to find pseudo-algebraic Ricci solitons on connected Lie groups.As applications, we prove that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and that any algebraic Ricci soliton on a semi-simple Lie group is Einstein. Furthermore, we construct several Lorentz algebraic Ricci solitons on the nilpotent Lie groups which have a codimension one abelian ideal.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0032","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43702906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-01DOI: 10.1515/advgeom-2021-0001
C. Atindogbe, M. Gutiérrez, R. Hounnonkpe
Abstract We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.
{"title":"Compact null hypersurfaces in Lorentzian manifolds","authors":"C. Atindogbe, M. Gutiérrez, R. Hounnonkpe","doi":"10.1515/advgeom-2021-0001","DOIUrl":"https://doi.org/10.1515/advgeom-2021-0001","url":null,"abstract":"Abstract We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2021-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47934123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-01DOI: 10.1515/advgeom-2020-0009
Sylwester Zając, Paweł Zapałowski
Abstract In this paper the complex geodesics of a convex domain in ℂn are studied. One of the main results provides a certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in ℂn. The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The ℂ-convexity of semitube domains is also discussed.
{"title":"Complex geodesics in convex domains and ℂ-convexity of semitube domains","authors":"Sylwester Zając, Paweł Zapałowski","doi":"10.1515/advgeom-2020-0009","DOIUrl":"https://doi.org/10.1515/advgeom-2020-0009","url":null,"abstract":"Abstract In this paper the complex geodesics of a convex domain in ℂn are studied. One of the main results provides a certain necessary condition for a holomorphic map to be a complex geodesic for a convex domain in ℂn. The established condition is of geometric nature and it allows to find a formula for every complex geodesic. The ℂ-convexity of semitube domains is also discussed.","PeriodicalId":7335,"journal":{"name":"Advances in Geometry","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/advgeom-2020-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45200436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：