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Journal articles on the topic 'Totally Umbilical'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

T., TSHIKUNA-MATAMBA. "A Note on Riemannian Submersions with Umbilical Fibres." Journal of Progressive Research in Mathematics 6, no. 2 (2016): 778–84. https://doi.org/10.5281/zenodo.3977521.

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In this paper, we discuss some geometric properties of Riemannian submersions whose fibres are totally contact umbilical. Some interrelations between totally contact umbilic, totally geodesic and minimality are established.

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2

Duggal, K. L., and D. H. Jin. "Totally umbilical lightlike submanifolds." Kodai Mathematical Journal 26, no. 1 (2003): 49–68. http://dx.doi.org/10.2996/kmj/1050496648.

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3

Devgan, Anu, and R. K. Nagaich. "Totally Contact Umbilical Radical Transversal Lightlike Submanifolds Of An Almost Contact Manifold With B-Metric." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 4 (2017): 7286–94. http://dx.doi.org/10.24297/jam.v13i4.6234.

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In the present paper, we study the geometry of totally contact umbilical radical transversal lightlike submanifolds and totally contact umbilical CR- submanifold of an indenite Sasaki-like almost contact manifold with B-metric. We nd the necessary and sucient condition for the characterization of the induced connection to be a metric connection. Finally, we have proved that for a totally contact umbilical CR-submanifold, totally contact umbilical radical transversal lightlike submanifold is a totally geodesic radical transversal lightlike submanifold.

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4

Sachdeva, Rashmi, Rakesh Kumar, and Satvinder Singh Bhatia. "Nonexistence of Totally Contact Umbilical Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds." ISRN Geometry 2013 (June 3, 2013): 1–8. http://dx.doi.org/10.1155/2013/231869.

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We study totally contact umbilical slant lightlike submanifolds of indefinite cosymplectic manifolds. We prove that there do not exist totally contact umbilical proper slant lightlike submanifolds in indefinite cosymplectic manifolds other than totally contact geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally we give characterization theorems on minimal slant lightlike submanifolds.

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5

Uddin, Siraj, Zafar Ahsan, and Yaakub Hadi. "Classification of totally umbilical slant submanifolds of a Kenmotsu manifold." Filomat 30, no. 9 (2016): 2405–12. http://dx.doi.org/10.2298/fil1609405u.

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The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold M of a Kenmotsu manifold ?M is either invariant or anti-invariant or dimM = 1 or the mean curvature vector H of M lies in the invariant normal subbundle. Moreover, we find with an example that every totally umbilical proper slant submanifold is totally geodesic.

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6

Schdeva, Rashmi, Rakesh Kumar, and Satvinder Singh Bhatia. "Totally contact umbilical slant lightlike submanifolds of indefinite Kenmotsu manifolds." Tamkang Journal of Mathematics 46, no. 2 (2015): 179–91. http://dx.doi.org/10.5556/j.tkjm.46.2015.1727.

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In this paper, we study totally contact umbilical slant lightlike submanifolds of indefinite Kenmotsu manifolds. We prove that there does not exist totally contact umbilical proper slant lightlike submanifold in indefinite Kenmotsu manifolds other than totally contact geodesic proper slant lightlike submanifold. We also prove that there does not exist totally contact umbilical proper slant lightlike submanifold of indefinite Kenmotsu space forms. Finally, we give some characterization theorems on minimal slant lightlike submanifolds of indefinite Kenmotsu manifolds.

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7

Rani, Rachna, Rakesh Kumar, and R. K. Nagaich. "Totally Contact Umbilical Lightlike Hypersurfaces of Indefinite -Manifolds." Journal of Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/395787.

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8

Al-Solamy, Falleh R., Meraj Ali Khan, and Siraj Uddin. "Totally Umbilical Hemi-Slant Submanifolds of Kaehler Manifolds." Abstract and Applied Analysis 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/987157.

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We study totally umbilical hemi-slant submanifolds of a Kaehler manifold via curvature tensor. We prove some classification theorems for totally umbilical hemi-slant submanifolds of a Kaehler manifold and give an example.

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9

Antić, Miroslava, Marilena Moruz, and Joeri Van der Veken. "H-Umbilical Lagrangian Submanifolds of the Nearly Kähler \( {\mathbb{S}^3\times\mathbb{S}^3} \)." Mathematics 8, no. 9 (2020): 1427. http://dx.doi.org/10.3390/math8091427.

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H-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that, in the homogeneous nearly Kähler S3×S3, also H-umbilical Lagrangian submanifolds are automatically totally geodesic.

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10

Nikolayevsky, Yury A. "Submanifolds with totally umbilical Gauss image." Geometriae Dedicata 62, no. 2 (1996): 115–38. http://dx.doi.org/10.1007/bf00147805.

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11

Uddin, Siraj, Meraj Ali Khan, and Khushwant Singh. "Totally Umbilical Proper Slant and Hemislant Submanifolds of an LP-Cosymplectic Manifold." Mathematical Problems in Engineering 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/516238.

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In the present note, we study slant and hemislant submanifolds of an LP-cosymplectic manifold which are totally umbilical. We prove that every totally umbilical proper slant submanifoldMof an LP-cosymplectic manifoldM¯is either totally geodesic or ifMis not totally geodesic inM¯then we derive a formula for slant angle ofM. Also, we obtain the integrability conditions of the distributions of a hemi-slant submanifold, and then we give a result on its classification.

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12

Siddesha, M. S., M. M. Praveena та C. S. Bagewadi. "A classification of totally umbilical proper slant and hemi-slant submanifolds of (k, μ)-contact manifolds". Journal of the Tensor Society 14, № 01 (2007): 30–39. http://dx.doi.org/10.56424/jts.v14i01.10608.

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The object of the present paper is to study slant and hemi-slant submanifolds of (k, μ)- contact manifolds which are totally umbilical. We prove that every totally umbilical proper slant submanifold M of a (k, μ)-contact manifold ˜M is either totally geodesic or if M is not totally geodesic then we derive a formula for slant angle. Also necessary and sufficient conditions for distributions of hemi-slant submanifolds to be integrable are worked out. Further we give a characterization theorem.

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13

Gök, Mustafa, and Erol Kılıç. "Totally umbilical semi-invariant submanifolds in locally decomposable metallic Riemannian manifolds." Filomat 36, no. 8 (2022): 2675–86. http://dx.doi.org/10.2298/fil2208675g.

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In this paper, we obtain some classification theorems for totally umbilical semi-invariant submanifolds in locally decomposable metallic Riemannian manifolds. We also prove that there exist no totally umbilical proper semi-invariant submanifolds in a posivitely or negatively curved locally decomposable metallic Riemannian manifold

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14

Kong, De-Xing, and Jinhua Wang. "Einstein's hyperbolic geometric flow." Journal of Hyperbolic Differential Equations 11, no. 02 (2014): 249–67. http://dx.doi.org/10.1142/s0219891614500076.

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We investigate the Einstein's hyperbolic geometric flow, which provides a natural tool to deform the shape of a manifold and to understand the wave character of metrics, the wave phenomenon of the curvature for evolutionary manifolds. For an initial manifold equipped with an Einstein metric and assumed to be a totally umbilical submanifold in the induced space-time, we prove that, along the Einstein's hyperbolic geometric flow, the metric is Einstein if and only if the corresponding manifold is a totally umbilical hypersurface in the induced space-time. For an initial manifold which is equippe

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15

Lee, Jae Won. "Characteristic Lightlike Submanifolds of an IndefiniteS-Manifold." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/140259.

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We study characteristicr-lightlike submanifoldsMtangent to the characteristic vector fields in an indefinite metricS-manifold, and we also discuss the existence of characteristic lightlike submanifolds of an indefiniteS-space form under suitable hypotheses: (1)Mis totally umbilical or (2) its screen distributionS(TM)is totally umbilical inM.

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16

HAIDER, S. M. KHURSHEED, V. A. KHAN, and S. I. HUSAIN. "TOTALLY UMBILICAL CR-SUBMANIFOLDS OF A KAEHLER MANIFOLD." Tamkang Journal of Mathematics 24, no. 1 (1993): 43–49. http://dx.doi.org/10.5556/j.tkjm.24.1993.4473.

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   In the present paper we study totally umbilical CR-submanifolds of a Kaehler manifold. A classification theorem for a $D^\perp$-totally umbilical CR-submanifold is proved. The conditions under which a CR- submanifold becomes a CR-product are obtained, and finally a theorem for a CR-submanifold to be a proper CR-product is also established.   

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17

Okumura, Masafumi, and Hiroshi Takahashi. "Non-immersibility of a space form as a totally umbilical hypersurface." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 109, no. 1-2 (1988): 17–21. http://dx.doi.org/10.1017/s0308210500026652.

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SynopsisSuppose that a space form is immersed into another Riemannian manifold as a totally umbilical hypersurface with constant mean curvature. Then, in the ambient manifold, the lengthof the curvature tensor, that of the Ricci tensor and the scalar curvature must satisfy an inequality. In this paper the authors proved the inequality. As applications of the inequality, some immersibility problems are investigated. For example, it is proved that if a space form is immersed in an Einstein manifold as a totally umbilical hypersurface, then the Einstein manifold has constant sectional curvature a

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18

Cheng, Qing-Ming, Haizhong Li, and Guoxin Wei. "The stability index of hypersurfaces with constant scalar curvature in spheres." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144, no. 3 (2014): 447–53. http://dx.doi.org/10.1017/s030821051200056x.

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The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stabili

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19

He, Qun, Wei Yang, and Wei Zhao. "On totally umbilical submanifolds of Finsler spaces." Annales Polonici Mathematici 100, no. 2 (2011): 147–57. http://dx.doi.org/10.4064/ap100-2-4.

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20

Stepanov, S. E., I. A. Alexandrova, I. I. Tsyganok, and J. Mikeš. "Conformal Killing Forms on Totally Umbilical Submanifolds." Journal of Mathematical Sciences 217, no. 5 (2016): 525–39. http://dx.doi.org/10.1007/s10958-016-2989-5.

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21

Sun, Huafei. "On totally umbilical submanifolds ofS n+p." Israel Journal of Mathematics 117, no. 1 (2000): 93–104. http://dx.doi.org/10.1007/bf02773565.

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22

Nikolaevskii, Yu A. "Totally umbilical submanifolds inG(2,n). II." Journal of Mathematical Sciences 72, no. 4 (1994): 3212–22. http://dx.doi.org/10.1007/bf01249521.

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23

Nikolaevskii, Yu A. "Totally umbilical manifolds inG(2,n). I." Journal of Mathematical Sciences 69, no. 1 (1994): 888–99. http://dx.doi.org/10.1007/bf01250820.

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24

Köprülü, Gizem, and Bayram Şahin. "Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers." International Journal of Geometric Methods in Modern Physics 18, no. 11 (2021): 2150169. http://dx.doi.org/10.1142/s0219887821501693.

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The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci c

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25

Banaru, M. B. "On a property of W4-manifolds." Differential Geometry of Manifolds of Figures, no. 51 (2020): 14–21. http://dx.doi.org/10.5922/0321-4796-2020-51-2.

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The properties of almost Hermitian manifolds belonging to the Gray — Hervella class W4 are considered. The almost Hermitian manifolds of this class were studied by such outstanding geometers like Alfred Gray, Izu Vaisman, and Vadim Feodorovich Kirichenko. Using the Cartan structural equations of an almost contact metric structure induced on an arbitrary oriented hypersurface of a W4-manifold, some results on totally umbilical and totally geodesic hypersurfaces of W4-manifolds are presented. It is proved that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a W4-manif

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26

ALI, SHAHID, and VIQAR AZAM KHAN. "TOTALLY UMBILICAL SUBMANIFOLDS OF A COMPLEX SPACE FORM." Tamkang Journal of Mathematics 22, no. 1 (1991): 105–6. http://dx.doi.org/10.5556/j.tkjm.22.1991.4579.

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27

KON, S. H., and SIN-LENG TAN. "TOTALLY UMBILICAL CR-SUBMANIFOLDS OF A NEARLY KAEHLER MANIFOLD." Tamkang Journal of Mathematics 27, no. 2 (1996): 145–49. http://dx.doi.org/10.5556/j.tkjm.27.1996.4352.

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The geometry of a CR-submanifold in a Kaehler manifold has been extensively studied. B.Y . Chen has classified the totally umbilical CR-submanifolds of a Kaehler manifold and showed that they are either totally geodesic, or totally real or dim$(D^{\perp}) =1$. In this paper we show that such a result is also true in a nearly Kaehler manifold.

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28

Aydın, Muhittin Evren, Adela Mihai, and Cihan Özgür. "Pythagorean Isoparametric Hypersurfaces in Riemannian and Lorentzian Space Forms." Axioms 11, no. 2 (2022): 59. http://dx.doi.org/10.3390/axioms11020059.

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We introduce the notion of a Pythagorean hypersurface immersed into an n+1-dimensional pseudo-Riemannian space form of constant sectional curvature c∈−1,0,1. By using this definition, we prove in Riemannian setting that if an isoparametric hypersurface is Pythagorean, then it is totally umbilical with sectional curvature φ+c, where φ is the Golden Ratio. We also extend this result to Lorentzian ambient space, observing the existence of a non totally umbilical model.

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29

de Lima, Henrique F., Antonio F. de Sousa, and Marco Antonio L. Velásquez. "Strong (r, s)-stability in the de Sitter space." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 1 (2015): 91–104. http://dx.doi.org/10.1017/s0308210513000346.

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In this paper, we establish the notion of strong (r, s)-stability concerning closed space-like hypersurfaces immersed with higher-order mean curvatures linearly related in the de Sitter space . In this setting, we prove that totally umbilical round spheres of are strongly (r, s)-stable. Afterwards, we obtain sufficient geometric conditions that guarantee that a closed strongly (r, s)-stable space-like hypersurface in must be a totally umbilical round sphere.

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30

Tondeur, Ph, and L. Vanhecke. "A characterisation of Riemannian foliations and totally umbilical submanifolds." Bulletin of the Australian Mathematical Society 48, no. 1 (1993): 101–8. http://dx.doi.org/10.1017/s0004972700015501.

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We discuss characterisations of Riemannian foliations, totally geodesic submanifolds, and totally umbilical submanifolds by sharp inequalities. These derive from the same linear algebraic set up, characterising a linear endomorphism which is a multiple of the identity.

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31

Şahin, Bayram. "Every Totally Umbilical Proper Slant Submanifold of a Kähler Manifold is Totally Geodesic." Results in Mathematics 54, no. 1-2 (2008): 167–72. http://dx.doi.org/10.1007/s00025-008-0324-2.

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32

Opozda, Barbara. "On totally umbilical submanifolds in nearly Kählerian manifolds." Annales Polonici Mathematici 49, no. 3 (1989): 221–27. http://dx.doi.org/10.4064/ap-49-3-221-227.

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33

Deszcz, Ryszard, Stanisław Ewert-Krzemieniewski, and Jerzy Policht. "On totally umbilical submanifolds of conformally birecurrent manifolds." Colloquium Mathematicum 55, no. 1 (1988): 79–96. http://dx.doi.org/10.4064/cm-55-1-79-96.

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34

Ewert-Krzemieniewski, Stanisław. "Totally umbilical submanifolds in some semi-Riemannian manifolds." Colloquium Mathematicum 119, no. 2 (2010): 269–99. http://dx.doi.org/10.4064/cm119-2-8.

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35

Massamba, Fortuné, and Samuel Ssekajja. "A new approach to totally umbilical null submanifolds." Novi Sad Journal of Mathematics 47, no. 2 (2017): 63–76. http://dx.doi.org/10.30755/nsjom.05090.

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36

Deshmukh, Sharief, and S. I. Husain. "Totally umbilical CR-submanifolds of a Kaehler manifold." Kodai Mathematical Journal 9, no. 3 (1986): 425–29. http://dx.doi.org/10.2996/kmj/1138037271.

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37

Laha, Barnali, and Arindam Bhattacharyya. "Totally umbilical hemislant submanifolds of LP-Sasakian manifold." Lobachevskii Journal of Mathematics 36, no. 2 (2015): 127–31. http://dx.doi.org/10.1134/s1995080215020122.

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38

Jin, Dae-Ho. "LIGHTLIKE REAL HYPERSURFACES WITH TOTALLY UMBILICAL SCREEN DISTRIBUTIONS." Communications of the Korean Mathematical Society 25, no. 3 (2010): 443–50. http://dx.doi.org/10.4134/ckms.2010.25.3.443.

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39

Dong, Junhong, and Ximin Liu. "Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes." ISRN Geometry 2014 (March 18, 2014): 1–10. http://dx.doi.org/10.1155/2014/974695.

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We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

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40

Bejancu, Aurel, and Hani Reda Farran. "On totally umbilical QR-submanifolds of quaternion Kaehlerian manifolds." Bulletin of the Australian Mathematical Society 62, no. 1 (2000): 95–103. http://dx.doi.org/10.1017/s0004972700018517.

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We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure.

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41

Kumar, Tejinder, Sangeet Kumar, and Pankaj Kumar. "Slant Lightlike Submanifolds of Semi-Riemannian Product Manifolds." Sarajevo Journal of Mathematics 19, no. 2 (2024): 155–70. http://dx.doi.org/10.5644/sjm.19.02.02.

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The aim of the present paper is to investigate geometric characteristics of slant lightlike submanifolds of semi-Riemannian product manifolds. We obtain characterization theorems for the existence of slant lightlike submanifolds of semi-Riemannian product manifolds. We also find a necessary and sufficient condition enabling the induced connection on slant lightlike submanifolds of semi-Riemannian product manifolds to be a metric connection. Then, we establish some results for the integrability of distributions associated with this class of lightlike submanifolds. Consequently, we investigate t

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42

Gadea, P. M., and A. Montesinos Amilibia. "Totally umbilical pseudo-Riemannian submanifolds of the paracomplex projective space." Czechoslovak Mathematical Journal 44, no. 4 (1994): 741–56. http://dx.doi.org/10.21136/cmj.1994.128493.

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43

Große, Nadine, and Roger Nakad. "Totally umbilical hypersurfaces of Spinc manifolds carrying special spinor fields." International Journal of Mathematics 31, no. 12 (2020): 2050100. http://dx.doi.org/10.1142/s0129167x20501001.

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Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin[Formula: see text] manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin[Formula: see text] case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to [Formula: see text] is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian [Formula: see text] manifolds of non-constant sectional curvature carrying

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44

Mao, Jing, and Shaodong Qin. "On pinching theorems for compact pseudo-umbilical submanifold." Demonstratio Mathematica 45, no. 3 (2012): 645–54. http://dx.doi.org/10.1515/dema-2013-0390.

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AbstractConsider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

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45

Bashir, M. A. "On the three-dimensionalCR-submanifolds of the six-dimensional sphere." International Journal of Mathematics and Mathematical Sciences 14, no. 4 (1991): 675–78. http://dx.doi.org/10.1155/s0161171291000893.

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46

Mohammed, Y. A. "Об обобщенных многообразиях Кенмоцу как гиперповерхностях многообразий Вайсмана - Грея". Владикавказский математический журнал 24, № 1 (2024): 5–12. http://dx.doi.org/10.46698/t2068-3621-5954-b.

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In this paper, we conclude that the hypersurfaces of Vaisman-Gray manifolds have generalized Kenmotsu structures under some conditions for the Lee form, Kirichenko's tensors and the second fundamental form of the immersion of the hypersurface into the manifold of Vaisman-Gray class. Moreover, the components of the second fundamental form are determined when the foregoing hypersurfaces have generalized Kenmotsu structures or any special kind of it or Kenmotsu structures, such that some of these components are vanish. Also, some components of Lee form and some components of some Kirichenko's ten

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47

Duggal, K. L. "Time-Dependent Evolving Null Horizons of a Dynamical Spacetime." ISRN Mathematical Physics 2014 (January 22, 2014): 1–10. http://dx.doi.org/10.1155/2014/291790.

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Totally geodesic null hypersurfaces have been widely used in the study of isolated black holes. In this paper, we introduce a new quasilocal notion of a family of totally umbilical null hypersurfaces called evolving null horizons (ENH) of a dynamical spacetime, satisfied under an appropriate energy condition. We focus on a variety of examples of ENHs and in some cases establish their relation with event and isolated horizons. We also present two specific physical models of an ENH in a black hole spacetime. Beside the examples, for further study we propose two open problems on possible general

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48

Karmakar, Payel. "Contact screen generic lightlike submanifolds of indefinite Kenmotsu manifold." Filomat 38, no. 26 (2024): 9167–84. https://doi.org/10.2298/fil2426167k.

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In this paper, we study contact screen generic lightlike (CSGL) submanifolds, totally umbilical CSGL submanifolds and minimal CSGL submanifolds of indefinite Kenmotsu manifolds. We investigate the necessary and sufficient (n & s) conditions for the induced connection on a CSGL submanifold to be a metric connection, for integrability & parallelism of some associated distributions, and for some distributions to be totally geodesic foliations. We also discuss about non-parallel distributions and more than one n & s conditions for a CSGL submanifold to be mixed geodesic. We further stu

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49

Banaru, G. A. "On quasi-Sasakian structure on a totally umbilical hypersurface of a six-dimensional Hermitian planar submanifold of Cayley algebra." Differential Geometry of Manifolds of Figures, no. 53 (2022): 5–12. http://dx.doi.org/10.5922/0321-4796-2022-53-1.

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Six-dimensional planar submanifolds of Cayley algebra equipped with almost Hermitian structures induced by Brown — Gray three-fold vector cross products in are considered. We select the case when the almost Hermitian structures on such six-dimensional planar submanifolds of Cayley algebra are Hermitian, i. e. these structures are integrable. We study almost contact metric structures on totally umbilical hypersurfaces in such six-dimensional Hermitian planar submanifolds of the octave algebra. We prove that if these almost contact metric structures on a totally umbilical hypersurface of a six-

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50

Piu, P., and M. Profir. "On the Three-Dimensional Homogenous SO(2)-Isotropic Riemannian Manifolds." Annals of the Alexandru Ioan Cuza University - Mathematics 57, no. 2 (2011): 361–76. http://dx.doi.org/10.2478/v10157-011-0032-1.

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Abstract:

On the Three-Dimensional Homogenous SO(2)-Isotropic Riemannian Manifolds In this paper we consider some properties of the three-dimensional homogenous SO(2)-isotropic Riemannian manifolds. In particular, we determine the geodesics, the totally geodesic surfaces, the totally umbilical surfaces and the geodesics of the rotational surfaces.

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