Pub Date : 2020-12-24DOI: 10.15673/tmgc.v13i3.1856
M. Zarichnyi, O. Polivoda
Haver's property C turns out to be related to Borst's transfinite extension of the covering dimension. We prove that, for a uncountably many countable ordinals β there exists a strongly universal kω-space for the class of spaces of transfinite covering dimension <β. In some sense, our result is a kω-counterpart of Radul's theorem on existence of absorbing sets of given transfinite covering dimension.
{"title":"Infinite-dimensional manifolds related to C-spaces","authors":"M. Zarichnyi, O. Polivoda","doi":"10.15673/tmgc.v13i3.1856","DOIUrl":"https://doi.org/10.15673/tmgc.v13i3.1856","url":null,"abstract":"Haver's property C turns out to be related to Borst's transfinite extension of the covering dimension. We prove that, for a uncountably many countable ordinals β there exists a strongly universal kω-space for the class of spaces of transfinite covering dimension <β. In some sense, our result is a kω-counterpart of Radul's theorem on existence of absorbing sets of given transfinite covering dimension.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88498096","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}
Pub Date : 2020-12-24DOI: 10.15673/tmgc.v13i4.1748
A. Pajitnov, Endo Hisaaki
This paper is about a generalization of celebrated Inoue's surfaces. To each matrix M in SL(2n+1,ℤ) we associate a complex non-Kähler manifold TM of complex dimension n+1. This manifold fibers over S1 with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.
{"title":"On the generalization of Inoue manifolds","authors":"A. Pajitnov, Endo Hisaaki","doi":"10.15673/tmgc.v13i4.1748","DOIUrl":"https://doi.org/10.15673/tmgc.v13i4.1748","url":null,"abstract":"This paper is about a generalization of celebrated Inoue's surfaces. To each matrix M in SL(2n+1,ℤ) we associate a complex non-Kähler manifold TM of complex dimension n+1. This manifold fibers over S1 with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds. \u0000 ","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84218500","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}
Pub Date : 2020-10-26DOI: 10.15673/tmgc.v14i2.1768
V. Babych, N. Golovashchuk
Applying geometric methods of 2-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem A is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering A', either A is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.
{"title":"Galois coverings of one-sided bimodule problems","authors":"V. Babych, N. Golovashchuk","doi":"10.15673/tmgc.v14i2.1768","DOIUrl":"https://doi.org/10.15673/tmgc.v14i2.1768","url":null,"abstract":"Applying geometric methods of 2-dimensional cell complex theory, we construct a Galois covering of a bimodule problem satisfying some structure, triangularity and finiteness conditions in order to describe the objects of finite representation type. Each admitted bimodule problem A is endowed with a quasi multiplicative basis. The main result shows that for a problem from the considered class having some finiteness restrictions and the schurian universal covering A', either A is schurian, or its basic bigraph contains a dotted loop, or it has a standard minimal non-schurian bimodule subproblem.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"133 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73247672","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}
Pub Date : 2020-10-13DOI: 10.15673/tmgc.v13i4.1750
Marek Golasinski
We review known and state some new results on homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of homotopy nilpotency of homogenous spaces G/K for a Lie group G and its closed subgroup K < G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)) and Ω(Vn,m(K)) of Grassmann Gn,m(K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is shown.
{"title":"A survey of homotopy nilpotency and co-nilpotency","authors":"Marek Golasinski","doi":"10.15673/tmgc.v13i4.1750","DOIUrl":"https://doi.org/10.15673/tmgc.v13i4.1750","url":null,"abstract":"We review known and state some new results on homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of homotopy nilpotency of homogenous spaces G/K for a Lie group G and its closed subgroup K < G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)) and Ω(Vn,m(K)) of Grassmann Gn,m(K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is shown.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"413 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79974971","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}
Pub Date : 2020-09-13DOI: 10.15673/TMGC.V13I2.1731
A. Prishlyak, M. V. Loseva
We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.
{"title":"Topology of optimal flows with collective dynamics on closed orientable surfaces","authors":"A. Prishlyak, M. V. Loseva","doi":"10.15673/TMGC.V13I2.1731","DOIUrl":"https://doi.org/10.15673/TMGC.V13I2.1731","url":null,"abstract":"We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86518968","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}
Pub Date : 2020-08-12DOI: 10.15673/tmgc.v13i2.1746
Claire David
In terms of analysis on fractals, the Sierpinski gasket stands out as one of the most studied example. The underlying aim of those studies is to determine a differential operator equivalent to the classic Laplacian. The classically adopted approach is a bidimensional one, through a sequence of so-called prefractals, i.e. a sequence of graphs that converges towards the considered domain. The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on the prefractals. It turns out that the gasket is also the image of a Peano curve, the so-called Arrowhead one, obtained by means of similarities from a starting point which is the unit line. This raises a question that appears of interest. Dirichlet forms solely depend on the topology of the domain, and not of its geometry. Which means that, if one aims at building a Laplacian on a fractal domain as the aforementioned curve, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account its specific geometry. Another difference due to the geometry, is encountered may one want to build a specific measure. For memory, the sub-cells of the Kigami and Strichartz approach are triangular and closed: the similarities at stake in the building of the Curve called for semi-closed trapezoids. As far as we know, and until now, such an approach is not a common one, and does not appear in such a context. It intererestingly happens that the measure we choose corresponds, in a sense, to the natural counting measure on the curve. Also, it is in perfect accordance with the one used in the Kigami and Strichartz approach. In doing so, we make the comparison -- and the link -- between three different approaches, that enable one to obtain the Laplacian on the arrowhead curve: the natural method; the Kigami and Strichartz approach, using decimation; the Mosco approach.
{"title":"Laplacian, on the Arrowhead Curve","authors":"Claire David","doi":"10.15673/tmgc.v13i2.1746","DOIUrl":"https://doi.org/10.15673/tmgc.v13i2.1746","url":null,"abstract":"\u0000In terms of analysis on fractals, the Sierpinski gasket stands out as one of the most studied example. \u0000The underlying aim of those studies is to determine a differential operator equivalent to the classic Laplacian. \u0000The classically adopted approach is a bidimensional one, through a sequence of so-called prefractals, i.e. a sequence of graphs that converges towards the considered domain. \u0000The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on the prefractals. \u0000 \u0000It turns out that the gasket is also the image of a Peano curve, the so-called Arrowhead one, obtained by means of similarities from a starting point which is the unit line. \u0000This raises a question that appears of interest. Dirichlet forms solely depend on the topology of the domain, and not of its geometry. \u0000Which means that, if one aims at building a Laplacian on a fractal domain as the aforementioned curve, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account its specific geometry. \u0000 \u0000Another difference due to the geometry, is encountered may one want to build a specific measure. \u0000For memory, the sub-cells of the Kigami and Strichartz approach are triangular and closed: the similarities at stake in the building of the Curve called for semi-closed trapezoids. \u0000As far as we know, and until now, such an approach is not a common one, and does not appear in such a context. \u0000 \u0000It intererestingly happens that the measure we choose corresponds, in a sense, to the natural counting measure on the curve. \u0000Also, it is in perfect accordance with the one used in the Kigami and Strichartz approach. \u0000In doing so, we make the comparison -- and the link -- between three different approaches, that enable one to obtain the Laplacian on the arrowhead curve: the natural method; the Kigami and Strichartz approach, using decimation; the Mosco approach. \u0000 \u0000 \u0000 ","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"PP 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84346848","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}
Pub Date : 2019-12-28DOI: 10.15673/tmgc.v12i4.1554
O. Artemovych, A. Balinsky, A. Prykarpatski
We review main differential-algebraic structures lying in background of analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and Leibniz type algebraic structures are derived, a new nonassociative "Riemann" algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. Subject to these important aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we revisited the classical Poisson manifold approach, closely related to our construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, we presented its natural and simple generalization allowing effectively to describe a wide class of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.
{"title":"Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras","authors":"O. Artemovych, A. Balinsky, A. Prykarpatski","doi":"10.15673/tmgc.v12i4.1554","DOIUrl":"https://doi.org/10.15673/tmgc.v12i4.1554","url":null,"abstract":"We review main differential-algebraic structures lying in background of analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and Leibniz type algebraic structures are derived, a new nonassociative \"Riemann\" algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. Subject to these important aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we revisited the classical Poisson manifold approach, closely related to our construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, we presented its natural and simple generalization allowing effectively to describe a wide class of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86525234","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}
Pub Date : 2019-10-19DOI: 10.15673/tmgc.v12i2.1485
Claire David
In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {mathcal W}(x)=displaystyle sum_{n=0}^{+infty} lambda^n,cos left(2, pi,N_b^n,x right)$, where $lambda$ and $N_b$ are two real numbers such that~mbox{$0 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.
{"title":"On fractal properties of Weierstrass-type functions","authors":"Claire David","doi":"10.15673/tmgc.v12i2.1485","DOIUrl":"https://doi.org/10.15673/tmgc.v12i2.1485","url":null,"abstract":"In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {mathcal W}(x)=displaystyle sum_{n=0}^{+infty} lambda^n,cos left(2, pi,N_b^n,x right)$, where $lambda$ and $N_b$ are two real numbers such that~mbox{$0 <lambda<1$},~mbox{$ N_b,in,N$} and $ lambda,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86620746","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}
Pub Date : 2019-10-15DOI: 10.15673/tmgc.v12i2.1455
Ярослав Виннишин, Віта Маркітан, Микола Вікторович Працьовитий, І. Г. Савченко
Наводиться конструкція континуальної сім'ї додатних рядів, множини неповних сум яких є канторвалами (об'єднанням ніде не щільної множини і множини, яка є нескінченним об'єднанням відрізків). Кожен ряд даної сім'ї має властивість $$sumlimits_{n=1}^{infty}a_{n}=1,~~~overline{lim_{nrightarrowinfty}}frac{a_n}{sum_{k=1}^{infty}a_{n+k}}=+infty,$$ причому для будь-якого $varepsilon>0$ в цій сім'ї існує ряд, міра Лебега множини неповних сум якого є більшою за $1-varepsilon$.
{"title":"Додатні ряди, множини підсум яких є канторвалами","authors":"Ярослав Виннишин, Віта Маркітан, Микола Вікторович Працьовитий, І. Г. Савченко","doi":"10.15673/tmgc.v12i2.1455","DOIUrl":"https://doi.org/10.15673/tmgc.v12i2.1455","url":null,"abstract":"Наводиться конструкція континуальної сім'ї додатних рядів, множини неповних сум яких є канторвалами (об'єднанням ніде не щільної множини і множини, яка є нескінченним об'єднанням відрізків). Кожен ряд даної сім'ї має властивість $$sumlimits_{n=1}^{infty}a_{n}=1,~~~overline{lim_{nrightarrowinfty}}frac{a_n}{sum_{k=1}^{infty}a_{n+k}}=+infty,$$ причому для будь-якого $varepsilon>0$ в цій сім'ї існує ряд, міра Лебега множини неповних сум якого є більшою за $1-varepsilon$.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85137801","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}
Pub Date : 2019-09-21DOI: 10.15673/tmgc.v12i2.1438
Koji Matsumoto
In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3 and the (CHR)3-scalar curvature τ3 in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.
{"title":"A (CHR)3-flat trans-Sasakian manifold","authors":"Koji Matsumoto","doi":"10.15673/tmgc.v12i2.1438","DOIUrl":"https://doi.org/10.15673/tmgc.v12i2.1438","url":null,"abstract":"In [4] M. Prvanovic considered several curvaturelike tensors \u0000defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3 and the (CHR)3-scalar curvature τ3 in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88311447","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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