Product submanifolds with pointwise 3-planar normal sections

Abstract

0. Introduction. Let M be a smooth m-dimensional submanifold in (m + d)-dimensional Euclidean space U m+d . For x e M and a non-zero vector X in T X M, we define theIn a neighbourhood of x, the intersection M DE(x,X) is a regular curve y : ( -e , e)-»M. We suppose the parameter f e (-e, e) is a multiple of the arc-length such that y(0) = x and y(0) = X. Each choice of X e T(M) yields a different curve which is called the normal section of M at JC in the direction of X, where X s T X (M) (Section 3).For such a normal section we can write Submanifolds with pointwise 3-planar normal sections have been studied by S. J. Li in the case when M is isotropic [6] and also in the case when M is spherical [7].In this paper we consider product submanifolds M = M X X M 2 with P3-PNS and we show that this implies strong conditions on Mj and M 2 .

Slant Curves in 3-Dimensional Almost Contact Metric Geometry

“…Now let a, b and ϕ be constants such that is an arclength parametrized curve in S 3 (see [30]). One can see that γ lies in the flat torus T 2 of constant mean curvature cot(2ϕ) given by the equations x 2 1 +y 2 1 = cos 2 ϕ and x 2 2 + y 2 2 = sin 2 ϕ. The tangent vector field T is T = (−a cos ϕ sin(as), a cos ϕ cos(as), −b sin ϕ sin(bs), b sin ϕ cos(bs)).…”

“…Hence we obtain Proposition 10.5. If γ is an almost Legendre curve in G(c 1 , c 2 ), then γ is ∇ tgeodesic if and only if the tangent vector field has the form γ ′ (s) = T 1 e 1 + T 2 e 2 , where T 1 and T 2 are constants satisfying T 2 1 + T 2 2 = 1.…”

On the Equiform Differential Geometry of AW(k)-Type Curves in Pseudo-Galilean 3-Space

“…The absolute of the pseudo-Galilean space is an ordered triple {w, f, I} where w is the ideal plane, f a line in w and I is the fixed hyperbolic involution of the points of f . In [4], from the differential geometric point of view, K. Arslan and A. West defined the notion of AW(k)-type submanifolds.…”

Classiffications of special curves in the three-dimensional Lie group

“…The notion of AW(k)-type submanifolds was introduced by Arslan and West in [2]. In particular, many works with AW(k)-type curves have been done by several authors.…”

“…[2]). Frenet curves of osculating order 3 in Lie group G with a bi-invariant metric are (i) of AW(1)-type if they satisfy N 3 (s) = 0, (ii) of AW(2)-type if they satisfy…”