In this article, we derive a Schur-type Inequality in terms of the gradient -Almost Newton-Ricci-Yamabe soliton in -contact metric manifolds. We discuss the triviality for the compact gradient -Almost Newton-Ricci-Yamabe soliton in -Contact metric manifolds. In the end, we deduce a Schur-type inequality for the gradient -Almost Newton-Yamabe soliton in -contact metric manifolds, static Riemannian manifolds, and normal homogeneous compact Riemannian manifolds coupled with a projected Casimir operator.
Citation: Mohd Danish Siddiqi, Fatemah Mofarreh. Schur-type inequality for solitonic hypersurfaces in -contact metric manifolds[J]. AIMS Mathematics, 2024, 9(12): 36069-36081. doi: 10.3934/math.20241711
| [1] | Erkan Oterkus, Timon Rabczuk, Selda Oterkus . Special Issue: Peridynamics and its applications. AIMS Materials Science, 2024, 11(3): 602-604. doi: 10.3934/matersci.2024030 |
| [2] | Mulugeta H. Woldemariam, Giovanni Belingardi, Ermias G. Koricho, Daniel T. Reda . Effects of nanomaterials and particles on mechanical properties and fracture toughness of composite materials: a short review. AIMS Materials Science, 2019, 6(6): 1191-1212. doi: 10.3934/matersci.2019.6.1191 |
| [3] | Haiming Wen . Preface to the special issue on advanced microstructural characterization of materials. AIMS Materials Science, 2016, 3(3): 1255-1255. doi: 10.3934/matersci.2016.3.1255 |
| [4] | Liang Wu . Cu-based mutlinary sulfide nanomaterials for photocatalytic applications. AIMS Materials Science, 2023, 10(5): 909-933. doi: 10.3934/matersci.2023049 |
| [5] | Kunio Hasegawa . Special issue on interaction of multiple cracks in materials—Volume 1 and 2. AIMS Materials Science, 2017, 4(2): 503-504. doi: 10.3934/matersci.2017.2.503 |
| [6] | Antonina M. Monaco, Anastasiya Moskalyuk, Jaroslaw Motylewski, Farnoosh Vahidpour, Andrew M. H. Ng, Kian Ping Loh, Milos Nesládek, Michele Giugliano . Coupling (reduced) Graphene Oxide to Mammalian Primary Cortical Neurons In Vitro. AIMS Materials Science, 2015, 2(3): 217-229. doi: 10.3934/matersci.2015.3.217 |
| [7] | Dimitra Papadaki, Spyros Foteinis, Vasileios Binas, Margarita N. Assimakopoulos, Theocharis Tsoutsos, George Kiriakidis . A life cycle assessment of PCM and VIP in warm Mediterranean climates and their introduction as a strategy to promote energy savings and mitigate carbon emissions. AIMS Materials Science, 2019, 6(6): 944-959. doi: 10.3934/matersci.2019.6.944 |
| [8] | Aum Rajpura, Hrutvik Prajapati, Anirban Sur, Vijaykumar S Jatti, Girish Kale, Yury Razoumny . Performance evaluation of lattice structured bumper beam for automobile. AIMS Materials Science, 2025, 12(3): 395-422. doi: 10.3934/matersci.2025021 |
| [9] | Ahmed Al-Ramthan, Ruaa Al Mezrakchi . Investigation of cementitious composites reinforced with metallic nanomaterials, boric acid, and lime for infrastructure enhancement. AIMS Materials Science, 2024, 11(3): 495-514. doi: 10.3934/matersci.2024025 |
| [10] | Nhung Thi-Tuyet Hoang, Anh Thi-Kim Tran, Nguyen Van Suc, The-Vinh Nguyen . Antibacterial activities of gel-derived Ag-TiO2-SiO2 nanomaterials under different light irradiation. AIMS Materials Science, 2016, 3(2): 339-348. doi: 10.3934/matersci.2016.2.339 |
In this article, we derive a Schur-type Inequality in terms of the gradient -Almost Newton-Ricci-Yamabe soliton in -contact metric manifolds. We discuss the triviality for the compact gradient -Almost Newton-Ricci-Yamabe soliton in -Contact metric manifolds. In the end, we deduce a Schur-type inequality for the gradient -Almost Newton-Yamabe soliton in -contact metric manifolds, static Riemannian manifolds, and normal homogeneous compact Riemannian manifolds coupled with a projected Casimir operator.
Due to a high surface to volume ratio, significant transport properties, and confinement effects resulting from the nanoscale dimensions, nanostructured materials are extensively studied for energy-related applications such as solar cells, catalysts, thermoelectrics, lithium ion batteries, super-capacitors, and hydrogen storage systems. Therefore extensive studies in this field are necessary to accomplish this challenge for the next decades. The editors of the Special Issue on Nanomaterials for energy and environmental applications in AIMS Materials Science tried to bring together experts in this field to describe the current state of the art of the study and to formulate new research directions. The Special issue contains eleven manuscripts from research groups in different countries presenting relevant results on energy related materials such as transparent conducting nanostructures (TiO2, SnO2, ZnO, Ga2O3, CuO, In2O3, GeO2, In2Ge2O7, ZnGa2O4, Zn2GeO4,Sb2O3),semiconductors (Si, Ge, CdSe, Ge3N4, InN) and organics. The contributors are at the forefront of the discipline, and their insightful scientific presentations will surely affect the next generations of materials scientists. The publications reflect the strong international interactions among physicists, chemists, engineers, materials scientists and technologists working in this field as well as the advantages, limitations and challenges of applications of nanomaterials. We hope that readers will find the papers interesting and stimulating further discussions in the field.
As guest editors of the special issue, we want to thank Dr. Shengjiao Pang, Editor and the technical staff who supported our initiative and made this special issue possible. Also, we thank all contributors of selected papers presented herein.
| [1] | C. De Lellis, P. Topping, Almost-Schur lemma, Calc. Var., 43 (2012), 347–354. http://dx.doi.org/10.1007/s00526-011-0413-z |
| [2] |
X. Cheng, A generalization of almost-Schur lemma for closed Riemannian manifolds, Ann. Glob. Anal. Geom., 43 (2013), 153–160. http://dx.doi.org/10.1007/s10455-012-9339-8 doi: 10.1007/s10455-012-9339-8
|
| [3] |
X. Cheng, An almost-Schur lemma for symmetric (0, 2) tensor and applications, Pac. J. Math., 267 (2014), 325–340. http://dx.doi.org/10.2140/pjm.2014.267.325 doi: 10.2140/pjm.2014.267.325
|
| [4] |
Y. Ge, G. Wang, An almost-Schur theorem on 4-dimensional manifolds, Proc. Amer. Math. Soc., 140 (2012), 1041–1044. http://dx.doi.org/10.1090/S0002-9939-2011-11065-7 doi: 10.1090/S0002-9939-2011-11065-7
|
| [5] |
E. Barbosa, A note on the almost-Schur lemma on 4-dimensional Riemannian closed manifold, Proc. Amer. Math. Soc., 140 (2012), 4319–4322. http://dx.doi.org/10.1090/S0002-9939-2012-11255-9 doi: 10.1090/S0002-9939-2012-11255-9
|
| [6] |
P. Ho, Almost Schur lemma for manifolds with boundary, Differ. Geom. Appl., 32 (2014), 97–112. http://dx.doi.org/10.1016/j.difgeo.2013.11.006 doi: 10.1016/j.difgeo.2013.11.006
|
| [7] | R. Hamilton, The Ricci flow on surfaces, Contemp. Math., 71 (1988), 237–261. |
| [8] |
S. Güler, M. Crasmareanu, Ricci-Yamabe maps for Riemannian flow and their volume variation and volume entropy, Turk. J. Math., 43 (2019), 2631–2641. http://dx.doi.org/10.3906/mat-1902-38 doi: 10.3906/mat-1902-38
|
| [9] |
M. D. Siddiqi, F. Mofarreh, M. A. Akyol, A. H. Hakami, -Ricci-Yamabe solitons along Riemannian submersions, Axioms, 12 (2023), 796. http://dx.doi.org/10.3390/axioms12080796 doi: 10.3390/axioms12080796
|
| [10] |
M. D. Siddiqi, U. De, S. Deshmukh, Estimation of almost Ricci-Yamabe solitons on static spacetimes, Filomat, 36 (2022), 397–407. http://dx.doi.org/10.2298/FIL2202397S doi: 10.2298/FIL2202397S
|
| [11] | S. Pigola, M. Rigoli, M. Rimoldi, A. Setti, Ricci almost solitons, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (2011), 757–799. http://dx.doi.org/10.2422/2036-2145.2011.4.01 |
| [12] |
A. W. Cunha, E. de Lima, H. de Lima, -almost Newton-Ricci solitons immersed into a Riemannian manifold, J. Math. Anal. Appl., 464 (2018), 546–556. http://dx.doi.org/10.1016/j.jmaa.2018.04.026 doi: 10.1016/j.jmaa.2018.04.026
|
| [13] | A. Cunha, E. de Lima, -almost Yamabe solitons in Lorentzian manifolds, Palestine Journal of Mathematics, 11 (2022), 521–530. |
| [14] |
U. C. De, M. D. Siddiqi, S. K. Chaubey, -almost Newton-Ricci solitons on Legenderian submanifolds of Sasakian space forms, Period. Math. Hung., 84 (2022), 76–88. http://dx.doi.org/10.1007/s10998-021-00394-x doi: 10.1007/s10998-021-00394-x
|
| [15] |
M. D. Siddiqi, Newton-Ricci Bourguignon almost solitons on Lagrangian submanifolds of complex space form, Acta Universitatis Apulensis, 63 (2020), 81–96. http://dx.doi.org/10.17114/j.aua.2020.63.07 doi: 10.17114/j.aua.2020.63.07
|
| [16] | M. D. Siddiqi, Ricci -soliton and geometrical structure in a dust fluid and viscous fluid sapcetime, Bulg. J. Phys., 46 (2019), 163–173. |
| [17] | M. D. Siddiqi, S. A. Siddiqui, M. Ahmad, -almost Newton-Yamabe solitons on submanifolds of Kenmotsu space forms, Palestine Journal of Mathematics, 11 (2022), 427–438. |
| [18] |
R. Sharma, Almost Ricci solitons and -contact geometry, Monatsh. Math., 175 (2014), 621–628. http://dx.doi.org/10.1007/s00605-014-0657-8 doi: 10.1007/s00605-014-0657-8
|
| [19] |
R. Sharma, Certain results on -contact and -contact manifolds, J. Geom., 89 (2008), 138–147. http://dx.doi.org/10.1007/s00022-008-2004-5 doi: 10.1007/s00022-008-2004-5
|
| [20] |
M. Tripathi, M. Dwaivedi, The structure of some classes of -contact manifolds, Proc. Math. Sci., 118 (2008), 371–379. http://dx.doi.org/10.1007/s12044-008-0029-1 doi: 10.1007/s12044-008-0029-1
|
| [21] | M. D. Siddiqi, Almost conformal Ricci solitons in -paracontact metric manifolds, Palestine Journal of Mathematics, 9 (2020), 832–840. |
| [22] |
S. Kaneyuki, F. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985), 173–187. http://dx.doi.org/10.1017/S0027763000021565 doi: 10.1017/S0027763000021565
|
| [23] |
S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom., 36 (2009), 37–60. http://dx.doi.org/10.1007/s10455-008-9147-3 doi: 10.1007/s10455-008-9147-3
|
| [24] |
D. Blair, Two remarks on contact metric structures, Tohoku Math. J., 29 (1977), 319–324. http://dx.doi.org/10.2748/tmj/1178240602 doi: 10.2748/tmj/1178240602
|
| [25] |
D. Blair, T. Koufogiorgos, R. Sharma, A classification of -dimensional contact metric manifolds with , Kodai Math. J., 13 (1990), 391–401. http://dx.doi.org/10.2996/kmj/1138039284 doi: 10.2996/kmj/1138039284
|
| [26] | B. Papantoniou, Contact Riemannian manifolds satisfying and -nullity distribution, Yokohama Mathematical Journal, 40 (1993), 149–161. |
| [27] |
D. Blair, T. Koufogiorgos, B. Papantoniou, Contact metric manifolds staisfying a nullity condition, Israel J. Math., 91 (1995), 189–214. http://dx.doi.org/10.1007/BF02761646 doi: 10.1007/BF02761646
|
| [28] |
E. Boeckx, A full classification of contact metric -spaces, Illinois J. Math., 44 (2000), 212–219. http://dx.doi.org/10.1215/ijm/1255984960 doi: 10.1215/ijm/1255984960
|
| [29] | H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 217–239. |
| [30] | S. Yau, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J., 25 (1976), 659–670. |
| [31] |
S. Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. IHES, 117 (2013), 247–269. http://dx.doi.org/10.1007/s10240-012-0047-5 doi: 10.1007/s10240-012-0047-5
|
| [32] | J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl., 58 (1979), 369–444. |
| [33] | S. Helgason, Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, Providence: American Mathematical Society, 1984. |
| [34] |
G. Lippner, D. Mangoubi, Z. McGuirk, R. Yovel, Strong convexity for harmonic functions on compact symmetric spaces, Proc. Amer. Math. Soc., 150 (2022), 1613–1622. http://dx.doi.org/10.1090/proc/15735 doi: 10.1090/proc/15735
|
Mohd Danish Siddiqi, Fatemah Mofarreh. Schur-type inequality for solitonic hypersurfaces in -contact metric manifolds[J]. AIMS Mathematics, 2024, 9(12): 36069-36081. doi: 10.3934/math.20241711
DownLoad: