Geometric properties of submanifolds of a Riemannian manifold in tangent bundles

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  • Mohammad Nazrul Islam Khan Qassim University
  • Nahid Fatima
  • Afifah Al Eid
  • B. B. Chaturvedi
  • Mohit Saxena


Tangent bundle,, Mathematical operators, Induced metric, Connection, Gauss equation, Weingarten equation, Codazzi equation, Curvature tensor, Hypersurface, Submanifold


The authors consider a quarter-symmetric semi-metric connection
(QSSM) connection in the tangent bundle and study the connection on
submanifold of co-dimension 2 and hypersurface concerning the QSSM
connection in the tangent bundle. Totally geodesic (TG), totally umbilical (TU), Gauss, Weingarten and Codazzi equations concerning the QSSM connection on submanifold of co-dimension 2 and hypersurface in the tangent bundle are obtained. Finally, we deduce Riemannian curvature tensor, Gauss and Codazzi equations on a submanifold of codimension 2 and hypersurface of Riemannian manifold concerning the quarter symmetric semi-metric connection in the tangent bundle.


S. Azami, General natural metallic structure on tangent bundle, Iran. J.

Sci. Technol. Trans. Sci. 42 (2018), 81-88.

N. S. Agashe and M. R. Chafle, A semi symetric non-metric connection in

a Riemannian manifold, Indian J. Pure Appl. Math. 23 (1992), 399-409.

O. Bahadir, Lorentzian para-Sasakian manifold with quarter-symmetric

non-metric connection, Journal of Dynamical Systems and Geometric Theories, 14(1) (2016), 17-33.

O. Bahadir, Lorenzian para-Sasakian manifold with quartersymmetric nonmetric connection, Journal of Dynamical Systems and Geometric Theories,

(1) (2016), 17-33.

O. Bahadir and S. K. Chaubey, Some notes on LP-Sasakian manifolds with

generalized symmetric metric connection, Honam Mathematical J. 42(3)

(2020), 461–476.

B. Barua, Submanifolds of a Riemannian manifold admitting a semisymmetric semi-metric connection, An. Stiint. Univ. Al. I. Cuza Iasi. Mat.

(1) (1998), 137-146.

M. Altunbas, Statistical structures and Killing vector fields on tangent bundles wrt two different metrics, Commun.Fac.Sci.Univ.Ank.Ser. A1 Math.

Stat., 72(3) (2023), 815–825.

Y. Li, A. Gezer and E. Karakas, Some notes on the tangent bundle with

a Ricci quarter-symmetric metric connection, AIMS Mathematics, 8(8)

(2023), 17335–17353.

A. Torun, M. Ozkan, ¨ Submanifolds of almost-complex metallic manifolds,

Mathematics, 11(5) (2023), 1172.

F. Yilmaz, M. Ozkan, ¨ On the generalized Gaussian fibonacci numbers and

Horadam hybrid numbers: A unified approach, Axioms, 11 (6) (2023), 255.

S. K. Chaubey and U. C. De, Characterization of the Lorentzian paraSasakian manifolds admitting a quarter-symmetric non-metric connection,

SUT Journal of Mathematics, 55(1) (2019), 53-67.

S. K. Chaubey and R. H. Ojha, On semi-symmetric non-metric and

quarter-symmetric metric connections, Tensor N. S. 70 (2008), 202-213.

S. K. Chaubey and U. C. De, Lorentzian para-Sasakian Manifolds Admitting a New type of Quarter-symmetric Non-metric ξ-connection, International Electronic Journal of Geometry, 12(1) (2019), 250-259.

A. Haseeb, S. K. Chaubey, F. Mofarreh, A. A. H. Ahmadini, A solitonic

sstudy of Riemannian manifolds equipped with a semi-symmetric metric

ξ-connection, Axioms 12(9) (2023), 809.

R. Kumar, L. Colney and M. N. I. Khan, Lifts of a semi-symmetric nonmetric connection (SSNMC) from statistical manifolds to the tangent bundle, Results in Nonlinear Analysis, 6(3) (2023), 50–65.

L. S. Das, R. Nivas and M. N. I. Khan, On submanifolds of codimension

immersed in a hsu–quarternion manifold, Acta Mathematica Academiae

Paedagogicae Nyiregyhaziensis, 25(1) (2009), 129-135.

U. C. De and D. Kamilya, hypersurfaces of Rieamnnian manifold with

semi-symmetric non-metric connection, J. Indian Inst. Sci. 75 (1995), 707-

A. Friedmann and J. A. Schouten, AUber die geometrie der halbsym- ¨

metrischen Aubertragung ¨ , Math. zeitschr. 21 (1924), 211-223.

K. Suwais, N. Ta¸s, N. Ozg¨ur and N. Mlaiki, ¨ Fixed Point Theorems in

symmetric Controlled M-Metric type Spaces, Symmetry, 15 (2023), 1665.

R. Qaralleh, A. Tallafha and W. Shatanawi, Some Fixed-Point Results

in Extended S-Metric Space of type (α, β), Symmetry, 15 (2023), 1790.

M .Rahim, K. Shah, T. Abdeljawad, M. Aphane, A. Alburaikan, and H.

A. E. W. Khalifa, Confidence levels-based p, q-quasirung orthopair fuzzy

operators and its applications to criteria group decision making problems,

IEEE Access, 1 (2023), 109983-109996. 10.1109/ACCESS.2023.3321876

S. Golab, On semi-symmetric and quarter-symmetric linear connections,

Tensor, N. S. 29 (1975), 249-254.

H. A. Hayden, Subspaces of a space with torsion, Proc.London Math. Soc.

(1932), 27-50.

S. K. Hui and R. S. Lemence, On Generalized ϕ-recurrent Kenmotsu

Manifolds with respect to Quarter-symmetric Metric Connection, KYUNGPOOK Math. J. 58(2018), 347-359.

M. N. I. Khan and L. S. Das, On CR-structures and the general quadratic

structure, Journal for Geometry and Graphics, 24(4)(2020), 249-255.

M. N. I. Khan, Liftings from a Para-Sasakian manifold to its tangent

bundles, FILOMAT, 37(20), 6727-6740, 2023.

M. N. I. Khan, Integrability of the metallic structures on the frame bundle,

Kyungpook Mathematical Journal, 61(4)2021, 791-803.

M. N. I. Khan, On Cauchy-Riemann structures and the general even order

structure, Journal of Science and Arts, 53(4) (2020), 801-808.

M. N. I. Khan and, L. S. Das, Parallelism of distributions and geodesics

on F(±a 2 , ±b 2 )-structure Lagrangian manifold, Facta Universitatis,

Series: Mathematics and Informatics, 36(1) (2021), 157-163.

M. N. I. Khan, Lifts of F(α, β)(3, 2, 1)-structures from manifolds to tangent bundles, Facta Universitatis, Series: Mathematics and Informatics 38

(1) (2023), 209-218.

M. N. I. Khan, Proposed theorems for lifts of the extended almost complex

structures on the complex manifold, Asian-European Journal of Mathematics, 15(11)(2022), 2250200.

M. N. I. Khan, F. Mofarreh and A. Haseeb, Tangent bundles of PSasakian manifolds endowed with a QSM connection, Symmetry, 15(3)

(2023), 753.

M. N. I. Khan, F. Mofarreh, A. Haseeb and M. Saxena, Certain results

on the lifts from an LP-Sasakian manifold to its tangent bundle associated

with a QSM connection, Symmetry, 15(8) (2023), 1553.

Y. Liang, On semi-symmetric recurrent-metric connection, Tensor N. S.

(1994), 107-112.

A. K. Mondal and U. C. De, Some properties of a quarter-symmetric metric connection on a sasakian manifold, Bulletin of Mathematical analysis

and applications, 1(2) (2009), 99-108.

M. Tani, Prolongations of hypersurfaces of tangent bundles, Kodai Math.

Semp. Rep. 21 (1969), 85-96.

K. Yano, S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker

Inc. New York, (1973).




How to Cite

Khan, M. N. I., Nahid Fatima, Al Eid, A., Chaturvedi, B. B., & Saxena, M. (2024). Geometric properties of submanifolds of a Riemannian manifold in tangent bundles. Results in Nonlinear Analysis, 7(2), 140–153. Retrieved from