• Communications of the Korean Mathematical Society
• /
• v.27 no.4
• /
• pp.763-770
• /
• 2012

In this paper, we study the geometry lightlike hypersurfaces (M, $g$, S(TM)) of a semi-Riemannian manifold ($\tilde{M}$, $\tilde{g}$) of quasi-constant curvature subject to the conditions: (1) The curvature vector field of $\tilde{M}$ is tangent to M, and (2) the screen distribution S(TM) is either totally geodesic in M or totally umbilical in $\tilde{M}$.

• Proceedings of the Korean Statistical Society Conference
• /
• 2003.05a
• /
• pp.37-42
• /
• 2003

We study the asymptotic expansion in small time of the mean distance of Brownian motion on Riemannian manifolds. We compute the first four terms of the asymptotic expansion of the mean distance by using the decomposition of Laplacian into homogeneous components. This expansion can he expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.4
• /
• pp.369-378
• /
• 2021

In this paper, we give an upper bound for the fundamental tone of stable constant mean curvature hypersurfaces in hyperbolic space. Let M be an n-dimensional complete non-compact constant mean curvature hypersurface with finite L²-norm of the traceless second fundamental form. If M is weakly stable, then λ₁(M) is bounded above by n² + O(n^2+s) for arbitrary s > 0.

• Journal of The Korean Society of Integrative Medicine
• /
• v.7 no.3
• /
• pp.149-157
• /
• 2019

Purpose : This study aims to check the relationship between the size of curvature in scoliosis patients and the reduction rate of curvature after wearing a brace and the relationships of the size of curvature and correction angle with Body Mass Index (BMI). Methods : With 30 adolescent girls who had never worn a brace, their Cobb angle and BMI were measured before manufacturing braces, and their corrected Cobb angle was measured after wearing the manufactured scoliosis braces. Results : The size of the major curvature before wearing the brace and the reduction rate of the curvature after putting it on had a negative correlation in both the major curvature (r=-.465, p<.01) and the minor curvature (r=-.516, p<.01). The size of the minor curvature and the reduction rate of the minor curvature before and after putting it on also had a negative correlation (r=-.429, p<.05). As for the relationship between the size of curvature and BMI, they had a negative correlation in both the major curvature (r=-.141) and the minor curvature (r=-.123), and as for the relationship between the reduction rate of curvature and BMI after wearing the brace, they had a positive correlation in both the major curvature (r=.251) and the minor curvature (r=.136); however, it was not statistically significant (p>.05). Conclusion : In conclusion, the larger the size of curvature, the lower the reduction rate of curvature after wearing the brace became. The larger the size of curvature, the lower the BMI became. The higher the BMI, the higher the correction ratio of the brace became. Therefore, it is judged that it is necessary to provide early treatment and manage body composition before scoliosis progresses.

• Bulletin of the Korean Mathematical Society
• /
• v.27 no.2
• /
• pp.133-142
• /
• 1990

In this paper, we define a new tensor field on a Sasqakian manifold, which is constructed from the conformal curvature tensor field by using the Boothby-Wang's fibration ([3]), and study some properties of this new tensor field. In Section 2, we recall definitions and fundamental properties of Sasakian manifold and .phi.-holomorphic sectional curvature. In Section 3, we define contact conformal curvature tensor field on a Sasakian manifold and prove that it is invariant under D-homothetic deformation due to S. Tanno([13]). In Section 4, we study Sasakian manifolds with vanishing contact conformal curvature tensor field, and the last Section 5 is devoted to studying some properties of fibred Riemannian spaces with Sasakian structure of vanishing contact conformal curvature tensor field.

• Korean Journal of Mathematics
• /
• v.7 no.2
• /
• pp.285-289
• /
• 1999

In this paper, we estimate the total curvature of non-parametric minimal surfaces by using the properties of univalent harmonic mappings defined on ${\Delta}=\{z:{\mid}z:{\mid}>1\}$.

• Communications of the Korean Mathematical Society
• /
• v.13 no.3
• /
• pp.539-544
• /
• 1998

Let (M, g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvature S, which is close to O. With conditions on a conformal invariant and scalar curvature of (M, g), we show that there exists a conformal metric (equation omitted), near g, whose scalar curvature (equation omitted) = 0 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_{i}$ with ∪$K_{i}$ = M.

• Journal of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.135-146
• /
• 2001

Let M(sup)n be a space-ike submanifold in a de Sitter space M(sub)p(sup)n+p (c) with constant scalar curvature. We firstly extend Cheng-Yau's Technique to higher codimensional cases. Then we study the rigidity problem for M(sup)n with parallel normalized mean curvature vector field.

• Honam Mathematical Journal
• /
• v.30 no.4
• /
• pp.677-683
• /
• 2008

On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker $s^*_g{{\neq}}0$ and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

• Journal of the Korean Mathematical Society
• /
• v.55 no.6
• /
• pp.1459-1468
• /
• 2018

We show that an umbilic-free minimal surface in ${\mathbb{R}}^3$ belongs to the associate family of the catenoid if and only if the geodesic curvatures of its lines of curvature have a constant ratio. As a corollary, the helicoid is shown to be the unique umbilic-free minimal surface whose lines of curvature have the same geodesic curvature. A similar characterization of the deformation family of minimal surfaces with planar lines of curvature is also given.