On the 5-dimensional Sasaki-Einstein manifold
|
|
- Clyde Nichols
- 5 years ago
- Views:
Transcription
1 Proceedings of The Fourteenth International Workshop on Diff. Geom. 14(2010) On the 5-dimensional Sasaki-Einstein manifold Byung Hak Kim Department of Applied Mathematics, Kyung Hee University, Suwon , Korea bhkim@khu.ac.kr (2010 Mathematics Subject Classification : 53C15, 53C25.) Abstract. In the talk, we will survey the characterizations and constructions of Sasaki- Einstein structure on the fibred Riemannian space. 1 Introduction A Sasakian manifold is a normal contact manifold. In some respects Sasakian manifold may be viewed as odd-dimensional analogue of Kaehler manifolds. Recently there are many papers with respect to the Sasakian geometry, in particular Sasaki-Einstein manifold. Sasaki-Einstein metric on odd-dimensional Riemannian manifold is deeply related to the Kaehler Ricci flat metric, that is Calabi-Yau metric on even-dimensional Riemannian manifold. More precisely Sasaki-Einstein manifold may be defined as an Einstein manifold whose metric cone is Ricci flat and Kaeher, that is Calabi-Yau manifold. Such manifolds provide interesting examples of the string theory [3,4,8]. In fact any complex surface whose metric is Kaehler-Einstein and of positive scalar curvature admits a unique simply connected circle bundle which is canonically Sasaki-Einstein. A classification of Riemannian manifolds admitting real Killing spinors on M correspond to the parallel spinors on C(M) = (R + M, dr 2 + r 2 g) the metric cone on M. In this point of a view, there are many results about Sasaki-Einstein geometry using this cone manifold with the Kaehler structure. Moreover, Sasaki Einstein metric is deeply related to the black hole theory, in particular dimensions 5 or 11. Fibred Riemannian space was first considered by Y. Muto [9] and treated by B. L. Reinhart [12] in the name of foliated Riemannian manifolds. B. O Neill [10] called such a foliation a Riemannian submersion and gave its structure equation. A. Gray [5] have studied pseudo-riemannian almost product manifolds and submersion. In the almost same time S. Ishihara and M. Konishi [6] developed an extensive theory of fibred Riemannian space. M. Ako [1] and T. Okubo [11] studied fibred space Key words and phrases: Sasaki-Einstein manifold, Fibred Riemannian space. 171
2 172 Byung Hak Kim with almost complex or almost Hermitian structure, and B. Watson [14] studied almost contact metric submersions. Y. Tashiro and B. H. Kim [13] have studied fibred Riemannian space with almost Hermitian or almost contact metric structure. In this point of a view, we will give a talk with respect to the construction of Sasaki-Einstein metric in the fibred Riemannian space and discuss about the characterizations of such spaces. 2 Fibred Riemannian space Let {M, B, G, π} be a fibred Riemannian space, that is M an m-dimensional total space with projectable Riemannian metric G, B an n-dimensional base space, and π : M B a projection with a maximal rank n. The fibre passing through a point q in M is denoted by F (q) or generally F, which is a p-dimensional submanifold of M, where p = m n. The quantities h and L are the components of the second fundamental tensor and normal connection of each fibre respectively. If the horizontal mapping covering curve in M is an isometry (resp. conformal mapping) of fibres, then it is called a fibred Riemannian space with isometric (resp. conformal) fibres. It is well known that a necessary and sufficient condition for M to have isomeric (resp. conformal) fibres is h = 0 (resp. h = λḡ, where ḡ is an induced Riemannian metric on each fibre). The following Theorem is well known [6]. Theorem 2.1. If the components of L and h vanish identically in a fibred Riemannian space, then the fibred space is locally the Riemannian product of the base space and a fibre. By Besse [2], the warped products of two Riemannian manifolds can be considered as a special case of a Riemannian submersion due to the following theorem. Theorem 2.2. Let M = B f 2 F be the warped product of (B, g) and (F, ḡ). Then the projection π : M B onto the first factor is a Riemannian submersion. More over the tensorial invariants of π satisfy ( ) L = 0, h = λḡ and the mean curvature vector is basic. Conversely, the conditions ( ) characterize locally warped products among Riemannain submersions.
3 On the 5-dimensional Sasaki-Einstein manifold Almost contact structure on the fibred Riemannian space There have been various attempts to clarify the relations between almost complex structures and almost contact structures. Using the fact that the structure groups of the tangent bundles of former and of the latter are reduced respectively to the unitary group. For an odd-dimensional manifold M 2n+1, A. Gray [5] defined an almost contact structure as a reduction of the structural group to U(n) 1. In terms of structure tensors we say M 2n+1 has an almost contact structure or sometimes (ϕ, ξ, η)-structure if M admits a tensor field ϕ of type (1, 1), a vector field ξ and a 1-form η satisfying (3.1) ϕ 2 = I + η ξ, η(ξ) = 1. It is well known that (3.1) reduce ϕξ = 0 and η ϕ = 0. If a manifold M 2n+1 with (ϕ, ξ, η)-structure admits a Riemannian metric g such that (3.2) g(ϕx, ϕy ) = g(x, Y ) η(x)η(y ), then we say M 2n+1 has an almost contact metric structure and g is called a compatible metric [3,6]. An almost contact structure (ϕ, ξ, η) on M is normal if the almost complex structure J on M R 1 given by J(X, f d dt ) = (ϕx fξ, η(x) d dt ), f being a C -function on M R 1, is integrable. An almost contact metric manifold (M, g) with (ϕ, ξ, η) is said to be [7,13] (i) contact if Φ = dη (ii) K-contact if Φ = dη and ξ is a Killing vector (iii) Sasakian if Φ = dη and (ϕ, ξ, η) is normal, where Φ(X, Y ) = g(ϕx, Y ). Y.Tashiro and B. H. Kim [13] have studied the fibred almost contact metric space with invariant fibres tangent to the structure vector and deal with various almost contact structure. They considered the fibred Riemannian space M with base space (B, g) with almost complex manifold with almost complex structure J and fibre F with almost contact structure ( ϕ, ξ, η, ḡ). If we put ϕ = J b a E b E a + ϕ β α C β C α, η = η α C α, ξ = ξ α C α, ( g 0 G = 0 ḡ ),
4 174 Byung Hak Kim then we can easily see that ( ϕ, ξ, η, G) defines an almost contact metric structure on M. Conversely, if there is in M an almost contact structure ( ϕ, ξ, η, G), G and ϕ are projectable and ξ is always vertical, then the structure induces an almost Hermitian structure (J, g) in the base space and almost contact metric structure ( ϕ, ξ, η, ḡ) in each fibre. In this case we have Theorem 3.1. [7] If a fibred almost contact metric structure is Sasakian, then the base space is Kaelerian and each fibre is Sasakian. In this case, each fibre is minimal, and L = J ξ, where J is a almost complex structure on the base space and ξ is a structure vector of the fibre. Theorem 3.2. [7] Let M be fibred Sasakian space with conformal fibres, then M is Sasaki-Einstein if and only if B is Kaeher-Einstein, S = λḡ n η η and K = n(n + 2p + K)/p, where S is a Ricci curvature tensor of the fibre and K is a scalar curvature of the fibre. In this case, each fibre is a totally geodesic submanifold of the total space and S = (α + 2)g, where α = K/m and K is a scalar curvature of the total space. Hence if we consider the 5-dimensional fibred Sasaki-Einstein space with conformal fibres, then we can characterize the geometric structure of the base space and each fibre in two cases, that is n = 4, p = 1 and n = 2, p = 3. Remark 3.3. The Sasakian manifold M can be considered as a cone manifold C(M) with Kaehler structure. Since the warped product is a special case of the fibred Riemanian space [2], we can reduced the properties of the Sasakian structure on M related to C(M) using the fundamental structure equation in the fibred Sasakian space with some conditions. Example 3.4. The Hopf fibration π : S 2n+1 CP n with fibre S 1 is fibred Sasaki- Einstein space with totally geodesic fibre. Obviously, CP n is Kaehler-Einstein. References [1] M. Ako Fibred space with almost complex structure, Kodai Math. Sem. Rep. 24 (1972), [2] A. Besse Einstein manifolds, Springer-Verlag, Berlin (1987).
5 On the 5-dimensional Sasaki-Einstein manifold 175 [3] C.P.Boyer and K.Galicki, On Sasaki-Einstein Geometry, Intl. J. Math. 11 (2000), arxiv:math. DG/ [4] Th. Friedrich and I. Kath, Einstein manifolds of dimension five with small first eigenvalue of the Dirac operator, J. Diff. Geom. 29 (1989), [5] A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), [6] S. Ishihara and M. Konishi, Differential geometry of fibred spaces, Publ. Study Group of Differential Geometry 7, Tokyo, (1973) [7] B. H. Kim, Fibred Riemannian space with contact structure, Hiroshima Math. J. 18 (1988), [8] J. M. Maldacena, The large N limit of superconformal field theories and super gravity, Adv. Theor. Math. Phys. 2 (1998), 231, arxiv:hep-th/ [9] Y. Muto, On some properties of a fibred Riemannian manifold Science reports of Yokohama Natl. Univ., bf 1 (1952), [10] B. O Neill, The fundamental equations of submersion, Michigan. Math. J., 13 (1966), [11] T. Okubo, Fibred spaces with almost Hermitian metrics whose base space admit almost contact metric structure, Math. Ann., 183 (1969), [12] B. L. Reinhart, Foliated manifolds with bundle like metrics, Ann. Math. 69 (1959), [13] Y. Tashiro and B. H. Kim, Almost complex and almost contact structures in fibred Riemannian space, Hiroshima Math. J., 18, Ser. A, (1988), [14] B. Watson, The differential geometry of two types of almost contact metric submersions, The Math. heritage of C. F. Gauss, (1991),
K. A. Khan, V. A. Khan and Sirajuddin. Abstract. B.Y. Chen [4] showed that there exists no proper warped CRsubmanifolds
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 21:2 (2007), 55 62 WARPED PRODUCT CONTACT CR-SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS
More informationON KENMOTSU MANIFOLDS
J. Korean Math. Soc. 42 (2005), No. 3, pp. 435 445 ON KENMOTSU MANIFOLDS Jae-Bok Jun, Uday Chand De, and Goutam Pathak Abstract. The purpose of this paper is to study a Kenmotsu manifold which is derived
More informationParallel and Killing Spinors on Spin c Manifolds. 1 Introduction. Andrei Moroianu 1
Parallel and Killing Spinors on Spin c Manifolds Andrei Moroianu Institut für reine Mathematik, Ziegelstr. 3a, 0099 Berlin, Germany E-mail: moroianu@mathematik.hu-berlin.de Abstract: We describe all simply
More informationComplex and real hypersurfaces of locally conformal Kähler manifolds
Complex and real hypersurfaces of locally conformal Kähler manifolds Odessa National Economic University Varna 2016 Topics 1 Preliminaries 2 Complex surfaces of LCK-manifolds 3 Real surfaces of LCK-manifolds
More informationA CHARACTERIZATION OF WARPED PRODUCT PSEUDO-SLANT SUBMANIFOLDS IN NEARLY COSYMPLECTIC MANIFOLDS
Journal of Mathematical Sciences: Advances and Applications Volume 46, 017, Pages 1-15 Available at http://scientificadvances.co.in DOI: http://dx.doi.org/10.1864/jmsaa_71001188 A CHARACTERIATION OF WARPED
More informationη = (e 1 (e 2 φ)) # = e 3
Research Statement My research interests lie in differential geometry and geometric analysis. My work has concentrated according to two themes. The first is the study of submanifolds of spaces with riemannian
More informationPOINTWISE SLANT SUBMERSIONS FROM KENMOTSU MANIFOLDS INTO RIEMANNIAN MANIFOLDS
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 2017 (561 572) 561 POINTWISE SLANT SUBMERSIONS FROM KENMOTSU MANIFOLDS INTO RIEMANNIAN MANIFOLDS Sushil Kumar Department of Mathematics Astronomy University
More informationarxiv:math/ v2 [math.dg] 25 May 2007
arxiv:math/0604008v2 [math.dg] 25 May 2007 A Note on Doubly Warped Product Contact CR-Submanifolds in trans-sasakian Manifolds Marian-Ioan Munteanu Abstract Warped product CR-submanifolds in Kählerian
More informationLegendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator
Note di Matematica 22, n. 1, 2003, 9 58. Legendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator Tooru Sasahara Department of Mathematics, Hokkaido University, Sapporo 060-0810,
More informationOn Einstein Nearly Kenmotsu Manifolds
International Journal of Mathematics Research. ISSN 0976-5840 Volume 8, Number 1 (2016), pp. 19-24 International Research Publication House http://www.irphouse.com On Einstein Nearly Kenmotsu Manifolds
More informationReduction of Homogeneous Riemannian structures
Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad
More informationA Joint Adventure in Sasakian and Kähler Geometry
A Joint Adventure in Sasakian and Kähler Geometry Charles Boyer and Christina Tønnesen-Friedman Geometry Seminar, University of Bath March, 2015 2 Kähler Geometry Let N be a smooth compact manifold of
More informationPseudoparallel Submanifolds of Kenmotsu Manifolds
Pseudoparallel Submanifolds of Kenmotsu Manifolds Sibel SULAR and Cihan ÖZGÜR Balıkesir University, Department of Mathematics, Balıkesir / TURKEY WORKSHOP ON CR and SASAKIAN GEOMETRY, 2009 LUXEMBOURG Contents
More informationWarped Product Bi-Slant Submanifolds of Cosymplectic Manifolds
Filomat 31:16 (2017) 5065 5071 https://doi.org/10.2298/fil1716065a Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: http://www.pmf.ni.ac.rs/filomat Warped Product
More informationON SEMI-INVARIANT SUBMANIFOLDS OF A NEARLY KENMOTSU MANIFOLD WITH THE CANONICAL SEMI-SYMMETRIC SEMI-METRIC CONNECTION. Mobin Ahmad. 1.
MATEMATIQKI VESNIK 62, 3 (2010), 189 198 September 2010 originalni nauqni rad research paper ON SEMI-INVARIANT SUBMANIFOLDS OF A NEARLY KENMOTSU MANIFOLD WITH THE CANONICAL SEMI-SYMMETRIC SEMI-METRIC CONNECTION
More informationReal hypersurfaces in a complex projective space with pseudo- D-parallel structure Jacobi operator
Proceedings of The Thirteenth International Workshop on Diff. Geom. 13(2009) 213-220 Real hypersurfaces in a complex projective space with pseudo- D-parallel structure Jacobi operator Hyunjin Lee Department
More informationAbstract. In this study we consider ϕ conformally flat, ϕ conharmonically. 1. Preliminaries
RADOVI MATEMATIČKI Vol. 12 (2003), 99 106 ϕ conformally flat Lorentzian para Sasakian manifolds (Turkey) Abstract. In this study we consider ϕ conformally flat, ϕ conharmonically flat and ϕ projectively
More informationGEOMETRY OF GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE IN TERMS OF THEIR GEODESICS
Mem. Gra. Sci. Eng. Shimane Univ. Series B: Mathematics 51 (2018), pp. 1 5 GEOMETRY OF GEODESIC SPHERES IN A COMPLEX PROJECTIVE SPACE IN TERMS OF THEIR GEODESICS SADAHIRO MAEDA Communicated by Toshihiro
More informationThe Schouten-van Kampen affine connections adapted to almost (para)contact metric structures (the work in progress) Zbigniew Olszak
The Schouten-van Kampen affine connections adapted to almost (para)contact metric structures (the work in progress) Zbigniew Olszak Wroc law University of Technology, Wroc law, Poland XVII Geometrical
More informationCANONICAL SASAKIAN METRICS
CANONICAL SASAKIAN METRICS CHARLES P. BOYER, KRZYSZTOF GALICKI, AND SANTIAGO R. SIMANCA Abstract. Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for
More informationGeometrical study of real hypersurfaces with differentials of structure tensor field in a Nonflat complex space form 1
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 9 (2018), pp. 1251 1257 Research India Publications http://www.ripublication.com/gjpam.htm Geometrical study of real hypersurfaces
More informationThe parallelism of shape operator related to the generalized Tanaka-Webster connection on real hypersurfaces in complex two-plane Grassmannians
Proceedings of The Fifteenth International Workshop on Diff. Geom. 15(2011) 183-196 The parallelism of shape operator related to the generalized Tanaka-Webster connection on real hypersurfaces in complex
More informationMathematical Research Letters 1, (1994) NEW EXAMPLES OF INHOMOGENEOUS EINSTEIN MANIFOLDS OF POSITIVE SCALAR CURVATURE
Mathematical Research Letters 1, 115 121 (1994) NEW EXAMPLES OF INHOMOGENEOUS EINSTEIN MANIFOLDS OF POSITIVE SCALAR CURVATURE Charles P. Boyer, Krzysztof Galicki, and Benjamin M. Mann Abstract. The purpose
More informationMEHMET AKIF AKYOL, LUIS M. FERNÁNDEZ, AND ALICIA PRIETO-MARTÍN
Konuralp Journal of Mathematics Volume No. 1 pp. 6 53 (016) c KJM THE L-SECTIONAL CURVATURE OF S-MANIFOLDS MEHMET AKIF AKYOL, LUIS M. FERNÁNDEZ, AND ALICIA PRIETO-MARTÍN Abstract. We investigate L-sectional
More informationCHAPTER 1 PRELIMINARIES
CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable
More informationDirac operators with torsion
Dirac operators with torsion Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Golden Sands / Bulgaria, September 2011 1 Relations between different objects on a Riemannian manifold (M n, g):
More informationSOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE. Toshiaki Adachi* and Sadahiro Maeda
Mem. Fac. Sci. Eng. Shimane Univ. Series B: Mathematical Science 32 (1999), pp. 1 8 SOME ASPECTS ON CIRCLES AND HELICES IN A COMPLEX PROJECTIVE SPACE Toshiaki Adachi* and Sadahiro Maeda (Received December
More informationH-projective structures and their applications
1 H-projective structures and their applications David M. J. Calderbank University of Bath Based largely on: Marburg, July 2012 hamiltonian 2-forms papers with Vestislav Apostolov (UQAM), Paul Gauduchon
More informationGeneralized almost paracontact structures
DOI: 10.1515/auom-2015-0004 An. Şt. Univ. Ovidius Constanţa Vol. 23(1),2015, 53 64 Generalized almost paracontact structures Adara M. Blaga and Cristian Ida Abstract The notion of generalized almost paracontact
More informationHard Lefschetz Theorem for Vaisman manifolds
Hard Lefschetz Theorem for Vaisman manifolds Antonio De Nicola CMUC, University of Coimbra, Portugal joint work with B. Cappelletti-Montano (Univ. Cagliari), J.C. Marrero (Univ. La Laguna) and I. Yudin
More informationEigenvalue estimates for Dirac operators with torsion II
Eigenvalue estimates for Dirac operators with torsion II Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Metz, March 2012 1 Relations between different objects on a Riemannian manifold (M n,
More informationSome Properties of a Semi-symmetric Non-metric Connection on a Sasakian Manifold
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 16, 789-799 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.28172 Some Properties of a Semi-symmetric Non-metric Connection on a Sasakian
More informationHolonomy groups. Thomas Leistner. School of Mathematical Sciences Colloquium University of Adelaide, May 7, /15
Holonomy groups Thomas Leistner School of Mathematical Sciences Colloquium University of Adelaide, May 7, 2010 1/15 The notion of holonomy groups is based on Parallel translation Let γ : [0, 1] R 2 be
More informationEinstein H-umbilical submanifolds with parallel mean curvatures in complex space forms
Proceedings of The Eighth International Workshop on Diff. Geom. 8(2004) 73-79 Einstein H-umbilical submanifolds with parallel mean curvatures in complex space forms Setsuo Nagai Department of Mathematics,
More informationReal Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivative
Canad. Math. Bull. Vol. 49 (1), 2006 pp. 134 143 Real Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivative Young Jin Suh Abstract. In this paper we give a characterization of
More informationAn inequality for warped product pseudo-slant submanifolds of nearly cosymplectic manifolds
Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 DOI 10.1186/s13660-015-0825-y R E S E A R C H Open Access An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic
More informationLINEAR CONNECTIONS ON NORMAL ALMOST CONTACT MANIFOLDS WITH NORDEN METRIC 1
LINEAR CONNECTIONS ON NORMAL ALMOST CONTACT MANIFOLDS WITH NORDEN METRIC 1 Marta Teofilova Abstract. Families of linear connections are constructed on almost contact manifolds with Norden metric. An analogous
More informationHolonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012
Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel
More information338 Jin Suk Pak and Yang Jae Shin 2. Preliminaries Let M be a( + )-dimensional almost contact metric manifold with an almost contact metric structure
Comm. Korean Math. Soc. 3(998), No. 2, pp. 337-343 A NOTE ON CONTACT CONFORMAL CURVATURE TENSOR Jin Suk Pak* and Yang Jae Shin Abstract. In this paper we show that every contact metric manifold with vanishing
More informationOn Indefinite Almost Paracontact Metric Manifold
International Mathematical Forum, Vol. 6, 2011, no. 22, 1071-1078 On Indefinite Almost Paracontact Metric Manifold K. P. Pandey Department of Applied Mathematics Madhav Proudyogiki Mahavidyalaya Bhopal,
More informationNotes on quasi contact metric manifolds
An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.) Tomul LXII, 016, f., vol. 1 Notes on quasi contact metric manifolds Y.D. Chai J.H. Kim J.H. Park K. Sekigawa W.M. Shin Received: 11.III.014 / Revised: 6.VI.014
More informationTwo simple ideas from calculus applied to Riemannian geometry
Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University
More informationDoubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds
INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY VOLUME 0 NO. 2 PAGE 73 8 207) Doubly Warped Products in Locally Conformal Almost Cosymplectic Manifolds Andreea Olteanu Communicated by Ion Miai) ABSTRACT Recently,
More informationHopf hypersurfaces in nonflat complex space forms
Proceedings of The Sixteenth International Workshop on Diff. Geom. 16(2012) 25-34 Hopf hypersurfaces in nonflat complex space forms Makoto Kimura Department of Mathematics, Ibaraki University, Mito, Ibaraki
More informationREAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WHOSE SHAPE OPERATOR IS OF CODAZZI TYPE IN GENERALIZED TANAKA-WEBSTER CONNECTION
Bull. Korean Math. Soc. 52 (2015), No. 1, pp. 57 68 http://dx.doi.org/10.4134/bkms.2015.52.1.057 REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WHOSE SHAPE OPERATOR IS OF CODAZZI TYPE IN GENERALIZED
More informationON AN EXTENDED CONTACT BOCHNER CURVATURE TENSOR ON CONTACT METRIC MANIFOLDS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXV 1993 FASC. 1 ON AN EXTENDED CONTACT BOCHNER CURVATURE TENSOR ON CONTACT METRIC MANIFOLDS BY HIROSHI E N D O (ICHIKAWA) 1. Introduction. On Sasakian
More informationReal Hypersurfaces with Pseudo-parallel Normal Jacobi Operator in Complex Two-Plane Grassmannians
Filomat 31:12 (2017), 3917 3923 https://doi.org/10.2298/fil1712917d Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Real Hypersurfaces
More informationJeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi
Bull. Korean Math. Soc. 40 (003), No. 3, pp. 411 43 B.-Y. CHEN INEQUALITIES FOR SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi Abstract. Some B.-Y.
More informationConservative Projective Curvature Tensor On Trans-sasakian Manifolds With Respect To Semi-symmetric Metric Connection
An. Şt. Univ. Ovidius Constanţa Vol. 15(2), 2007, 5 18 Conservative Projective Curvature Tensor On Trans-sasakian Manifolds With Respect To Semi-symmetric Metric Connection C.S.Bagewadi, D.G.Prakasha and
More informationContact manifolds and generalized complex structures
Contact manifolds and generalized complex structures David Iglesias-Ponte and Aïssa Wade Department of Mathematics, The Pennsylvania State University University Park, PA 16802. e-mail: iglesias@math.psu.edu
More informationDifferential Geometry of Warped Product. and Submanifolds. Bang-Yen Chen. Differential Geometry of Warped Product Manifolds. and Submanifolds.
Differential Geometry of Warped Product Manifolds and Submanifolds A warped product manifold is a Riemannian or pseudo- Riemannian manifold whose metric tensor can be decomposes into a Cartesian product
More informationHow to recognize a conformally Kähler metric
How to recognize a conformally Kähler metric Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:0901.2261, Mathematical Proceedings of
More informationTRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap
TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold
More informationQing-Ming Cheng and Young Jin Suh
J. Korean Math. Soc. 43 (2006), No. 1, pp. 147 157 MAXIMAL SPACE-LIKE HYPERSURFACES IN H 4 1 ( 1) WITH ZERO GAUSS-KRONECKER CURVATURE Qing-Ming Cheng and Young Jin Suh Abstract. In this paper, we study
More informationAn Inequality for Warped Product Semi-Invariant Submanifolds of a Normal Paracontact Metric Manifold
Filomat 31:19 (2017), 6233 620 https://doi.org/10.2298/fil1719233a Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat An Inequality for
More informationContact Metric Manifold Admitting Semi-Symmetric Metric Connection
International Journal of Mathematics Research. ISSN 0976-5840 Volume 6, Number 1 (2014), pp. 37-43 International Research Publication House http://www.irphouse.com Contact Metric Manifold Admitting Semi-Symmetric
More informationON ϕ-pseudo SYMMETRIC KENMOTSU MANIFOLDS Shyamal Kumar Hui 1
Novi Sad J. Math. Vol. 43, No. 1, 2013, 89-98 ON ϕ-pseudo SYMMETRIC KENMOTSU MANIFOLDS Shyamal Kumar Hui 1 Abstract. The object of the present paper is to study ϕ-pseudo symmetric and ϕ-pseudo Ricci symmetric
More informationSCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KENMOTSU MANIFOLDS
SARAJEVO JOURNAL OF MATHEMATICS Vol.7 (19) (2011), 103 113 SCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KENMOTSU MANIFOLDS RAM SHANKAR GUPTA AND A. SHARFUDDIN Abstract. In this paper, we introduce
More informationLet F be a foliation of dimension p and codimension q on a smooth manifold of dimension n.
Trends in Mathematics Information Center for Mathematical Sciences Volume 5, Number 2,December 2002, Pages 59 64 VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS KYOUNG HEE HAN AND HOBUM KIM Abstract.
More informationTHREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE
THREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE BENJAMIN SCHMIDT AND JON WOLFSON ABSTRACT. A Riemannian manifold has CVC(ɛ) if its sectional curvatures satisfy sec ε or sec ε pointwise, and if every tangent
More informationSome Research Themes of Aristide Sanini. 27 giugno 2008 Politecnico di Torino
Some Research Themes of Aristide Sanini 27 giugno 2008 Politecnico di Torino 1 Research themes: 60!s: projective-differential geometry 70!s: Finsler spaces 70-80!s: geometry of foliations 80-90!s: harmonic
More informationGENERALIZED VECTOR CROSS PRODUCTS AND KILLING FORMS ON NEGATIVELY CURVED MANIFOLDS. 1. Introduction
GENERALIZED VECTOR CROSS PRODUCTS AND KILLING FORMS ON NEGATIVELY CURVED MANIFOLDS Abstract. Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we
More informationOn some special vector fields
On some special vector fields Iulia Hirică Abstract We introduce the notion of F -distinguished vector fields in a deformation algebra, where F is a (1, 1)-tensor field. The aim of this paper is to study
More informationOn para-norden metric connections
On para-norden metric connections C. Ida, A. Manea Dedicated to Professor Constantin Udrişte at his 75-th anniversary Abstract. The aim of this paper is the construction of some para-norden metric connections
More informationC-parallel Mean Curvature Vector Fields along Slant Curves in Sasakian 3-manifolds
KYUNGPOOK Math. J. 52(2012), 49-59 http://dx.doi.org/10.5666/kmj.2012.52.1.49 C-parallel Mean Curvature Vector Fields along Slant Curves in Sasakian 3-manifolds Ji-Eun Lee Institute of Mathematical Sciences,
More informationBulletin of the Transilvania University of Braşov Vol 6(55), No Series III: Mathematics, Informatics, Physics, 9-22
Bulletin of the Transilvania University of Braşov Vol 6(55), No. 1-013 Series III: Mathematics, Informatics, Physics, 9- CONHARMONIC CURVATURE TENSOR ON KENMOTSU MANIFOLDS Krishnendu DE 1 and Uday Chand
More informationAlmost Kenmotsu 3-h-manifolds with cyclic-parallel Ricci tensor
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 4206 4213 Research Article Almost Kenmotsu 3-h-manifolds with cyclic-parallel Ricci tensor Wenjie Wang Henan Engineering Laboratory for
More informationarxiv: v3 [math.dg] 13 Mar 2011
GENERALIZED QUASI EINSTEIN MANIFOLDS WITH HARMONIC WEYL TENSOR GIOVANNI CATINO arxiv:02.5405v3 [math.dg] 3 Mar 20 Abstract. In this paper we introduce the notion of generalized quasi Einstein manifold,
More informationKilling fields of constant length on homogeneous Riemannian manifolds
Killing fields of constant length on homogeneous Riemannian manifolds Southern Mathematical Institute VSC RAS Poland, Bedlewo, 21 October 2015 1 Introduction 2 3 4 Introduction Killing vector fields (simply
More informationLAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS
LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere
More informationNew Einstein Metrics on 8#(S 2 S 3 )
New Einstein Metrics on 8#(S 2 S 3 ) Charles P. Boyer Krzysztof Galicki Abstract: We show that #8(S 2 S 3 ) admits two 8-dimensional complex families of inequivalent non-regular Sasakian-Einstein structures.
More informationTHE SASAKI CONE AND EXTREMAL SASAKIAN METRICS
THE SASAKI CONE AND EXTREMAL SASAKIAN METRICS CHARLES P. BOYER, KRZYSZTOF GALICKI, AND SANTIAGO R. SIMANCA Abstract. We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold.
More informationRiemannian submersions and eigenforms of the Witten Laplacian
Proceedings of The Sixteenth International Workshop on Diff. Geom. 16(2012) 143-153 Riemannian submersions and eigenforms of the Witten Laplacian Hyunsuk Kang Department of Mathematics, Korea Institute
More informationOn a Type of Para-Kenmotsu Manifold
Pure Mathematical Sciences, Vol. 2, 2013, no. 4, 165-170 HIKARI Ltd, www.m-hikari.com On a Type of Para-Kenmotsu Manifold T. Satyanarayana Department of Mathematics Pragati Engineering College, Surampalem,
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationOn homogeneous Randers spaces with Douglas or naturally reductive metrics
On homogeneous Randers spaces with Douglas or naturally reductive metrics Mansour Aghasi and Mehri Nasehi Abstract. In [4] Božek has introduced a class of solvable Lie groups with arbitrary odd dimension.
More informationLagrangian Submanifolds with Constant Angle Functions in the Nearly Kähler S 3 S 3
Lagrangian Submanifolds with Constant Angle Functions in the Nearly Kähler S 3 S 3 Burcu Bektaş Istanbul Technical University, Istanbul, Turkey Joint work with Marilena Moruz (Université de Valenciennes,
More informationConformal Killing forms on Riemannian manifolds
Conformal Killing forms on Riemannian manifolds Habilitationsschrift zur Feststellung der Lehrbefähigung für das Fachgebiet Mathematik in der Fakultät für Mathematik und Informatik der Ludwig-Maximilians-Universität
More informationRigidity and Non-rigidity Results on the Sphere
Rigidity and Non-rigidity Results on the Sphere Fengbo Hang Xiaodong Wang Department of Mathematics Michigan State University Oct., 00 1 Introduction It is a simple consequence of the maximum principle
More informationarxiv: v1 [math.dg] 30 Jul 2016
SASAKI-EINSTEIN 7-MANIFOLDS, ORLIK POLYNOMIALS AND HOMOLOGY RALPH R. GOMEZ arxiv:1608.00564v1 [math.dg] 30 Jul 2016 Abstract. Let L f be a link of an isolated hypersurface singularity defined by a weighted
More informationComplete integrability of geodesic motion in Sasaki-Einstein toric spaces
Complete integrability of geodesic motion in Sasaki-Einstein toric spaces Mihai Visinescu Department of Theoretical Physics National Institute for Physics and Nuclear Engineering Horia Hulubei Bucharest,
More informationSCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE SASAKIAN MANIFOLDS
An. Şt. Univ. Ovidius Constanţa Vol. 18(2), 2010, 315 336 SCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE SASAKIAN MANIFOLDS Cumali Yıldırım, Bayram Ṣahin Abstract We introduce screen transversal
More informationTheorem 2. Let n 0 3 be a given integer. is rigid in the sense of Guillemin, so are all the spaces ḠR n,n, with n n 0.
This monograph is motivated by a fundamental rigidity problem in Riemannian geometry: determine whether the metric of a given Riemannian symmetric space of compact type can be characterized by means of
More informationAFFINE SPHERES AND KÄHLER-EINSTEIN METRICS. metric makes sense under projective coordinate changes. See e.g. [10]. Form a cone (1) C = s>0
AFFINE SPHERES AND KÄHLER-EINSTEIN METRICS JOHN C. LOFTIN 1. Introduction In this note, we introduce a straightforward correspondence between some natural affine Kähler metrics on convex cones and natural
More informationAdapted complex structures and Riemannian homogeneous spaces
ANNALES POLONICI MATHEMATICI LXX (1998) Adapted complex structures and Riemannian homogeneous spaces by Róbert Szőke (Budapest) Abstract. We prove that every compact, normal Riemannian homogeneous manifold
More informationHARMONIC MAPS AND PARA-SASAKIAN GEOMETRY. S. K. Srivastava and K. Srivastava. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 3 2017, 153 163 September 2017 research paper originalni nauqni rad HARMONIC MAPS AND PARA-SASAKIAN GEOMETRY S. K. Srivastava and K. Srivastava Abstract. The purpose
More informationarxiv: v1 [math.dg] 28 Aug 2014
LOCAL CLASSIFICATION AND EXAMPLES OF AN IMPORTANT CLASS OF PARACONTACT METRIC MANIFOLDS VERÓNICA MARTÍN-MOLINA arxiv:1408.6784v1 [math.dg] 28 Aug 2014 Abstract. We study paracontact metric (κ,µ)-spaces
More informationSome results on K-contact and Trans-Sasakian Manifolds
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No., 008, (1-31) ISSN 1307-5543 www.ejpam.com Some results on K-contact and Trans-Sasakian Manifolds Bagewadi Channabasappa 1,, Basavarajappa N.S,
More informationContact pairs (bicontact manifolds)
Contact pairs (bicontact manifolds) Gianluca Bande Università degli Studi di Cagliari XVII Geometrical Seminar, Zlatibor 6 September 2012 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds)
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationRICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS
J. Austral. Math. Soc. 72 (2002), 27 256 RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS ION MIHAI (Received 5 June 2000; revised 19 February 2001) Communicated by K. Wysocki Abstract Recently,
More informationSUBTANGENT-LIKE STATISTICAL MANIFOLDS. 1. Introduction
SUBTANGENT-LIKE STATISTICAL MANIFOLDS A. M. BLAGA Abstract. Subtangent-like statistical manifolds are introduced and characterization theorems for them are given. The special case when the conjugate connections
More information1. Introduction In the same way like the Ricci solitons generate self-similar solutions to Ricci flow
Kragujevac Journal of Mathematics Volume 4) 018), Pages 9 37. ON GRADIENT η-einstein SOLITONS A. M. BLAGA 1 Abstract. If the potential vector field of an η-einstein soliton is of gradient type, using Bochner
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), 79 7 www.emis.de/journals ISSN 176-0091 WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS ADELA MIHAI Abstract. B.Y. Chen
More informationGeneralized Contact Structures
Generalized Contact Structures Y. S. Poon, UC Riverside June 17, 2009 Kähler and Sasakian Geometry in Rome in collaboration with Aissa Wade Table of Contents 1 Lie bialgebroids and Deformations 2 Generalized
More informationclass # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS
class # 34477 MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS [DG] stands for Differential Geometry at https://people.math.osu.edu/derdzinski.1/courses/851-852-notes.pdf [DFT]
More informationFrom holonomy reductions of Cartan geometries to geometric compactifications
From holonomy reductions of Cartan geometries to geometric compactifications 1 University of Vienna Faculty of Mathematics Berlin, November 11, 2016 1 supported by project P27072 N25 of the Austrian Science
More informationKILLING TENSORS AND EINSTEIN WEYL GEOMETRY
C O L L O Q U I U M M A T H E M A T I C U M VOL. 81 1999 NO. 1 KILLING TENSORS AND EINSTEIN WEYL GEOMETRY BY W LODZIMIERZ J E L O N E K (KRAKÓW) Abstract. We give a description of compact Einstein Weyl
More informationarxiv: v1 [math.dg] 25 Dec 2018 SANTIAGO R. SIMANCA
CANONICAL ISOMETRIC EMBEDDINGS OF PROJECTIVE SPACES INTO SPHERES arxiv:82.073v [math.dg] 25 Dec 208 SANTIAGO R. SIMANCA Abstract. We define inductively isometric embeddings of and P n (C) (with their canonical
More informationHermitian vs. Riemannian Geometry
Hermitian vs. Riemannian Geometry Gabe Khan 1 1 Department of Mathematics The Ohio State University GSCAGT, May 2016 Outline of the talk Complex curves Background definitions What happens if the metric
More information