# On Some Geometric Properties of the Sphere $S^n$

Volume 2, Issue 4, pp 607--618
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### Authors

Richard S. Lemence - Institute of Mathematics, College of Science, University of the Philippines, Diliman, Quezon City, Philippines Dennis T. Leyson - Institute of Mathematics, College of Science, University of the Philippines, Diliman, Quezon City, Philippines Marian P. Roque - Institute of Mathematics, College of Science, University of the Philippines, Diliman, Quezon City, Philippines

### Abstract

It is known that the sphere $S^n$ admits an almost complex structure only when $n = 2$ or $n = 6$ . In this paper, we show that the sphere $S^n$ is a space of constant sectional curvature and using the results of T. Sato in [4], we determine the scalar curvature and the *-scalar curvature of $S^6$. We shall also prove that $S^6$ is a non-Kähler nearly Kähler manifold using the Levi-Civita connection on $S^6$ defined by H. Hashimoto and K. Sekigawa [3]. In [2], A. Gray and L. Hervella defined sixteen classes of almost Hermitian manifolds. We shall define quasi-Hermitian, a class of almost Hermitian manifolds and partially characterize almost Hermitian manifolds that belong to this class. Finally, under certain conditions, we shall show the sphere $S^6$ is quasi-Hermitian.

### Share and Cite

##### ISRP Style

Richard S. Lemence, Dennis T. Leyson, Marian P. Roque, On Some Geometric Properties of the Sphere $S^n$, Journal of Mathematics and Computer Science, 2 (2011), no. 4, 607--618

##### AMA Style

Lemence Richard S., Leyson Dennis T., Roque Marian P., On Some Geometric Properties of the Sphere $S^n$. J Math Comput SCI-JM. (2011); 2(4):607--618

##### Chicago/Turabian Style

Lemence, Richard S., Leyson, Dennis T., Roque, Marian P.. "On Some Geometric Properties of the Sphere $S^n$." Journal of Mathematics and Computer Science, 2, no. 4 (2011): 607--618

### Keywords

• Sphere
• Kähler manifolds
• Hermitian manifolds
• quasi- Hermitian manifolds

•  53B35
•  53C55

### References

• [1] A. Newlander, L. Nirenberg , Complex Analytic Coordinates in Almost Complex Manifolds , Annals of Mathematics, 2 (1957), 391--404

• [2] A. Gray, L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Annali di Matematica, 123 (1980), 35--58

• [3] H. Hashimoto, K. Sekigawa, Submanifolds of a nearly-Kähler 6- dimensional sphere, Proceedings of the Eighth International Workshop on Differential Geometry, 2004 (2004), 23--45

• [4] T. Sato, Curvatures of almost Hermitian manifolds, Graduate School of Science and Technology (Doctoral Thesis), 1--84 (1992)

• [5] B. S. Kruglikov , Nijenhuis tensors and obstructions for pseudoholomorphic mapping constructions, University of Tromso, 1996 (1996), 43 pages