• 대한수학회보
• /
• 제51권5호
• /
• 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.

• 대한수학회지
• /
• 제41권6호
• /
• pp.1035-1047
• /
• 2004

In this paper, we study the unit tangent sphere bundles T$_1$M(4c) of complex space forms M(4c) with constant holomorphic sectional curvature 4c. In particular, we determine T$_1$M(4c) whose Ricci tensors satisfy the Einstein-like conditions.

• 호남수학학술지
• /
• 제46권1호
• /
• pp.38-46
• /
• 2024

Let X, Y be Banach spaces, S_X and S_Y be the unit sphere of X and Y, respectively. Let f₀ : S_X → S_Y be ε-isometry for some ε ≥ 0. In this paper, we show that there is an extension f : X → Y of f₀ such that f is linear.

• 대한수학회지
• /
• 제46권6호
• /
• 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.

• 충청수학회지
• /
• 제13권2호
• /
• pp.41-51
• /
• 2001

We show that an n-dimensional unit sphere in the 2n-dimensional de Sitter space can be associated to a solution of a partial differential equation on a Lorentzian Grassmannian system coming from the integrable system theory.

• 호남수학학술지
• /
• 제22권1호
• /
• pp.83-90
• /
• 2000

We establish a sufficient condition for a simple closed curve in the Euclidean plain $\mathbb{R}^2$, the unit sphere $S^2$ or the hyperbolic plane $H^2$ to be the boundary of a metric ball.

• 대한수학회보
• /
• 제48권3호
• /
• pp.585-600
• /
• 2011

In this paper we derive an integral formula on an (n + 3)-dimensional, compact, minimal contact three CR-submanifold M of (p-1) contact three CR-dimension immersed in a unit (4m+3)-sphere $S^{4m+3}$. Using this integral formula, we give a sufficient condition concerning the scalar curvature of M in order that such a submanifold M is to be a generalized Clifford torus.

• 대한수학회보
• /
• 제53권6호
• /
• pp.1715-1723
• /
• 2016

We study the characteristic Jacobi operator ${\ell}={\bar{R}({\cdot},{\xi}){\xi}$ (along the Reeb flow ${\xi}$) on the unit tangent sphere bundle $T_1M$ over a Riemannian manifold ($M^n$, g). We prove that if ${\ell}$ is pseudo-parallel, i.e., ${\bar{R}{\cdot}{\ell}=L{\mathcal{Q}}({\bar{g}},{\ell})$, by a non-positive function L, then M is locally flat. Moreover, when L is a constant and $n{\neq}16$, M is of constant curvature 0 or 1.

• 대한수학회보
• /
• 제42권4호
• /
• pp.777-787
• /
• 2005

In this paper we study (n + 1)-dimensional compact contact CR-submanifolds of (n - 1) contact CR-dimension immersed in an odd-dimensional unit sphere $S^{2m+1}$. Especially we provide necessary conditions in order for such a sub manifold to be the generalized Clifford surface $$S^{2n_1+1}(((2n_1+1)/(n+1))^{\frac{1}{2}})\;{\times}\;S^{2n_2+1}(((2n_2+1)/(n+1)^{\frac{1}{2}})$$ for some portion (n1, n2) of (n - 1)/2 in terms with sectional curvature.

• 호남수학학술지
• /
• 제36권4호
• /
• pp.805-812
• /
• 2014

For unit tangent sphere bundles $T_1M$ with the standard contact metric structure (${\eta},\bar{g},{\phi},{\xi}$), we have two fundamental operators that is, $h=\frac{1}{2}{\pounds}_{\xi}{\phi}$ and ${\ell}=\bar{R}({\cdot},{\xi}){\xi}$, where ${\pounds}_{\xi}$ denotes Lie differentiation for the Reeb vector field ${\xi}$ and $\bar{R}$ denotes the Riemmannian curvature tensor of $T_1M$. In this paper, we study the Reeb ow invariancy of the corresponding (0, 2)-tensor fields H and L of h and ${\ell}$, respectively.