• 대한수학회보
• /
• 제46권2호
• /
• pp.255-261
• /
• 2009

In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M, g) := (SU(2), g) with an arbitrary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ${||X||}_g$ = 1,X ${\in}$ X(M)}, where Ric is the Ricci tensor field on (M, g), and then get a necessary and sufficient condition for the Levi-Civita connection ${\nabla}$ on the manifold (M, g) to be projectively flat. Furthermore, we obtain a necessary and sufficient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제14권4호
• /
• pp.231-253
• /
• 2007

The purpose of this paper is to study the geometry of null curves in a semi-Riemannian manifold (M, g) of index 2. We show that it is possible to construct new Frenet equations of two types of null curves in M.

• Kyungpook Mathematical Journal
• /
• 제53권4호
• /
• pp.603-614
• /
• 2013

In the article we show that nondieomorphic symplectic 4-manifolds which admit marked Lefschetz fibrations can share the same monodromy group. Explicitly we prove that, for each integer g > 0, every knot surgery 4-manifold in a family {$E(2)_K{\mid}K$ is a bered 2-bridge knot of genus g in $S^3$} admits a marked Lefschetz fibration structure which has the same monodromy group.

• 호남수학학술지
• /
• 제41권2호
• /
• pp.369-380
• /
• 2019

Inspired by the problem of finding hyperelastic curves in a Riemannian manifold, we present a study on the variational problem of a hyperelastic curve in Lie group. In a Riemannian manifold, we reorganize the characterization of the hyperelastic curve with appropriate constraints. By using this equilibrium equation, we derive an Euler-Lagrange equation for the hyperelastic energy functional defined in a Lie group G equipped with bi-invariant Riemannian metric. Then, we give a solution of this equation for a null hyperelastic Lie quadratic when Lie group G is SO(3).

• 대한수학회논문집
• /
• 제34권1호
• /
• pp.279-286
• /
• 2019

Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the $Pl{\ddot{u}}cker$ embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$ where $N=\(^n_k\)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2{\leq}k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$. We show that, for the Sullivan model ${\phi}:A{\rightarrow}B$, where A and B are the Sullivan minimal models of ${\mathbb{C}}P^{N-1}$ and $G_{n,k}$ respectively, the evaluation subgroup $G_n(A,B;{\phi})$ of ${\phi}$ is generated by a single element and the relative evaluation subgroup $G^{rel}_n(A,B;{\phi})$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.

• 한국분무공학회지
• /
• 제1권1호
• /
• pp.28-34
• /
• 1996

Study on the mechanism of droplet transport and the droplet eddy diffusivity in the intake manifold of internal conbustion engine with carburetor has been carried out in this paper The theory and experiments were studied and performed respectively, to elucidate the mechanism and to measure typical rates of deposition, on the walls of a straight type intake manifold, of water droplets suspended in a turbulent air streams. Accordingly, the results are that Mechanism of a spray transport to the walls is caused by the fluctuation component of radial velocity. Deposition rate of a spray on the walls is mainly dependent upon air velocity and mean diameter of spray, and Droplet eddy diffusivity in the intake manifold is around $80\sim105cm^2/sec$.

• 대한수학회논문집
• /
• 제36권4호
• /
• pp.817-827
• /
• 2021

In this paper, we prove that a diffeomorphism f on a normal almost contact 3-manifold M is a CRL-transformation if and only if M is an α-Sasakian manifold. Moreover, we show that a CR-loxodrome in an α-Sasakian 3-manifold is a pseudo-Hermitian magnetic curve with a strength $q={\tilde{r}}{\eta}({\gamma}^{\prime})=(r+{\alpha}-t){\eta}({\gamma}^{\prime})$ for constant 𝜂(𝛄'). A non-geodesic CR-loxodrome is a non-Legendre slant helix. Next, we prove that let M be an α-Sasakian 3-manifold such that (∇_YS)X = 0 for vector fields Y to be orthogonal to ξ, then the Ricci tensor 𝜌 satisfies 𝜌 = 2α²g. Moreover, using the CRL-transformation $\tilde{\nabla}^t$ we fine the pseudo-Hermitian curvature $\tilde{R}$, the pseudo-Ricci tensor $\tilde{\rho}$ and the torsion tensor field $\tilde{T}^t(\tilde{S}X,Y)$.

• 대한수학회논문집
• /
• 제23권2호
• /
• pp.257-268
• /
• 2008

Let G and K be compact subgroups of orthogonal groups and $0{\leq}r<x<{\infty}$. We prove that every topological fiber bundle over a definable $C^r$ manifold whose structure group is K admits a unique strongly definable $C^r$ fiber bundle structure up to definable $C^r$ fiber bundle isomorphism. We prove that every G vector bundle over an affine definable $C^rG$ manifold admits a unique strongly definable $C^rG$ vector bundle structure up to definable $C^rG$ vector bundle isomorphism.

• 충청수학회지
• /
• 제23권3호
• /
• pp.495-510
• /
• 2010

An Einstein's connection which takes the form (2.23) is called a $^*g$-ME-connection and the corresponding vector is called a $^*g$-ME-vector. The $^*g$-ME-manifold is a generalized n-dimensional Riemannian manifold $X_n$ on which the differential geometric structure is imposed by the unified field tensor $^*g^{{\lambda}{\nu}}$, satisfying certain conditions, through the $^*g$-ME-connection and we denote it by $^*g-MEX_n$. The purpose of this paper is to derive a general representation and a special representation of the $^*g$-ME-vector in $^*g-MEX_n$.

• 대한수학회보
• /
• 제56권5호
• /
• pp.1297-1314
• /
• 2019

In this work, we investigate the following fractional boundary value problems $$\{_tD^{\alpha}_T({\mid}_0D^{\alpha}_t(u(t)){\mid}^{p-2}_0D^{\alpha}_tu(t))\\={\nabla}W(t,u(t))+{\lambda}g(t){\mid}u(t){\mid}^{q-2}u(t),\;t{\in}(0,T),\\u(0)=u(T)=0,$$ where ${\nabla}W(t,u)$ is the gradient of W(t, u) at u and $W{\in}C([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ is homogeneous of degree r, ${\lambda}$ is a positive parameter, $g{\in}C([0,T])$, 1 < r < p < q and ${\frac{1}{p}}<{\alpha}<1$. Using the Fibering map and Nehari manifold, for some positive constant ${\lambda}_0$ such that $0<{\lambda}<{\lambda}_0$, we prove the existence of at least two non-trivial solutions