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Complete classification of surfaces with a canonical principal direction in the Euclidean space $$ \mathbb{E} $$ 3

100
Munteanu M., Nistor A.
Open Mathematics
|
2011
|
tom 9
|
nr 2
378-389
EN
In the present paper we classify all surfaces in $$ \mathbb{E} $$ 3 with a canonical principal direction. Examples of this type of surfaces are constructed. We prove that the only minimal surface with a canonical principal direction in the Euclidean space $$ \mathbb{E} $$ 3 is the
catenoid.
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2
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On an inequality of Oprea for Lagrangian submanifolds

100
Dillen F., Fastenakels J.
Open Mathematics
|
2009
|
tom 7
|
nr 1
140-144
EN
We show that a Lagrangian submanifold of a complex space form attaining equality in the inequality obtained by Oprea in [8], must be totally geodesic.
3
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Quasi-minimal rotational surfaces in pseudo-Euclidean four-dimensional space

81
Ganchev G., Milousheva V.
Open Mathematics
|
2014
|
tom 12
|
nr 10
1586-1601
EN
In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In
this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.
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4
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Polynomial translation surfaces of Weingarten types in Euclidean 3-space

81
Yoon D.
Open Mathematics
|
2010
|
tom 8
|
nr 3
430-436
EN
In this paper, we classify polynomial translation surfaces in Euclidean 3-space satisfying the Jacobi condition with respect to the Gaussian curvature, the mean curvature and the second Gaussian curvature.
5
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On canonical screen for lightlike submanifolds of codimension two

62
Duggal K.
Open Mathematics
|
2007
|
tom 5
|
nr 4
710-719
EN
In this paper we study two classes of lightlike submanifolds of codimension two of semi-Riemannian manifolds, according as their radical subspaces are 1-dimensional or 2-dimensional. For a large variety of both these classes, we prove the existence of integrable canonical screen distributions
subject to some reasonable geometric conditions and support the results through examples.
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6
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On geometry of curves of flags of constant type

52
Doubrov B., Zelenko I.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1836-1871
EN
We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one
can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.
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