• Journal for History of Mathematics
• /
• v.24 no.3
• /
• pp.23-35
• /
• 2011

The sphere theorem is one of the main streams in modern Riemannian geometry. In this article, we survey developments of pinching theorems from the classical one to the recent differentiable pinching theorem. Also we include sphere theorems of metric invariants such as diameter and radius with historical view point.

• Journal of the Society of Naval Architects of Korea
• /
• v.54 no.3
• /
• pp.250-257
• /
• 2017

It is desirable to have a way to predict the pressure drag due to various appendages attached to stern. As a mathematical model for these, a sphere and a singularity behind it, both in the uniform flow can be considered. We may use the Butler's sphere theorem to find the Stokes' stream function when the resulting flow is axisymmetric, and then the extended Lagally's theorem to get the force upon the sphere due to the singularity. Assuming the separation distance between the sphere and the singularity is small, the leading order approximation for the force is obtained and it is found out that if the separation distance and the square root of the strength of the dipole are of the same order, the effect of the image of the dipole with respect to the sphere is the most important.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.12 no.4
• /
• pp.239-247
• /
• 2008

Let $I{\in}C^{1,1}$ be a strongly indefinite functional defined on a Hilbert space H. We investigate the number of the critical points of I when I satisfies two pairs of Torus-Sphere variational linking inequalities and when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities. We show that I has at least four critical points when I satisfies two pairs of Torus-Sphere variational linking inequality with $(P.S.)^*_c$ condition. Moreover we show that I has at least 2m critical points when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities with $(P.S.)^*_c$ condition. We prove these results by Theorem 2.2 (Theorem 1.1 in [1]) and the critical point theory on the manifold with boundary.

• Bulletin of the Society of Naval Architects of Korea
• /
• v.20 no.4
• /
• pp.33-40
• /
• 1983

The drift force acting on a freely-floating sphere in water of finite depth is studied within the framework of a linear potential theory. A velocity potential describing fluid motion is determined by distribution pulsating sources and dipoles on the immersed surface of the sphere. Upon knowing values of the potential, hydrodynamic forces are evaluated by integrating pressures over the immersed surface of the sphere. The motion response of the sphere in water of finite depth is obtained by solving the equation of motion. From these results, the drift force on the sphere is evaluated by the momentum theorem, in which a far-field velocity potential is utilized in forms of Kochin function. The drift force coefficient Cdr of a fixed sphere increases monotononically with non-dimensional wave frequency ${\sigma}a$. On the other hand, in freely-floating case, the Cdr has a peak value at ${\sigma}a$ of heave resonance. The magnitude of the drift force coefficient Cdr in the case of finite depth is different form that for deep water, but the general tendency seems to be similar in both cases. It is to note that Cdr is greater than 1.0 when non-dimensional water depth d/a is 1.5 in the case of freely-floating sphere.

• Journal of the Korean Mathematical Society
• /
• v.46 no.6
• /
• pp.1255-1265
• /
• 2009

We show that the Riemannian geometry of a tangent sphere bundle of a Riemannian manifold (M, g) of constant radius $\gamma$ reduces essentially to the one of unit tangent sphere bundle of a Riemannian manifold equipped with the respective induced Sasaki metrics. Further, we provide some applications of this theorem on the $\eta$-Einstein tangent sphere bundles and certain related topics to the tangent sphere bundles.

• Journal of the Korean Mathematical Society
• /
• v.48 no.2
• /
• pp.329-340
• /
• 2011

We shall give some curvature conditions for the unit tangent sphere bundle of an n($\geq$ 4)-dimensional Riemannian manifold to be H-contact. Furthermore, we provide an example illustrating Main Theorem.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.5
• /
• pp.1375-1397
• /
• 2014

In this paper we establish codimension reduction theorem for submanifolds of a (4m+3)-dimensional unit sphere $S^{4m+3}$ with Sasakian 3-structure and apply it to submanifolds of a quaternionic projective space.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.613-618
• /
• 1996

In this paper, M will always denote an n- dimensional complete Riemannian manifold and $\omega_n$ is the volume of $S^n$.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.385-392
• /
• 1996

Smale [16] posed the following question; is having an attracting periodic orbit a generic property for diffeomorphisms of two-sphere $S^2\ulcorner$(A generic property of $f \in Diff(M)$ is one that is true for a Baire set in Diff(M)). Mane[5] and Plykin[13] had an positive answer for Axiom A diffeomorphisms of $S^2$. To explain our theorem, we begin by briefly recalling stability conjecture posed by palis and smale.

• Communications of the Korean Mathematical Society
• /
• v.13 no.1
• /
• pp.95-107
• /
• 1998

In this paper we prove the Hopf-Rinow theorem for CAT(0) spaces and show that the ideal boundaries of complete CAT(0) manifolds of dimension 2 or 3 with some additional conditions are homeomorphic to the circle or 2-sphere by the characterization of the local shadows around the branch points.