2018
DOI: 10.3390/math6080130
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Abstract: In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the Laplace operator, Δ, on the submanifold, it satisfies ΔG=fG+gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we cl… Show more

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Cited by 7 publications
(12 citation statements)
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“…from which ε 3 c 1 Q + c 2 Q = 0, a contradiction. Therefore, we get: Combining Definition 3, Theorems 2 and 3, and the classification theorem of flat surfaces with the generalized 1-type Gauss map in Minkowski 3-space in [8], we have the following: Theorem 4. Let M be a non-cylindrical ruled surface of type M 1 + , M 3 + , or M 1 − in E 3 1 with the generalized 1-type Gauss map.…”
Section: Non-cylindrical Ruled Surfaces With the Generalized 1-type Gmentioning
confidence: 97%
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“…from which ε 3 c 1 Q + c 2 Q = 0, a contradiction. Therefore, we get: Combining Definition 3, Theorems 2 and 3, and the classification theorem of flat surfaces with the generalized 1-type Gauss map in Minkowski 3-space in [8], we have the following: Theorem 4. Let M be a non-cylindrical ruled surface of type M 1 + , M 3 + , or M 1 − in E 3 1 with the generalized 1-type Gauss map.…”
Section: Non-cylindrical Ruled Surfaces With the Generalized 1-type Gmentioning
confidence: 97%
“…Now, we suppose that φ = 0. From (8), we see that the functions f and g depend only on the parameter s, i.e., f (s, t) = f (s) and g(s, t) = g(s). Taking the derivative of Equation (9) and using (10), we get:…”
Section: Cylindrical Ruled Surfaces In E 3 1 With the Generalized 1-tmentioning
confidence: 99%
“…Since the ruled surface M is non-cylindrical, M is one of an open part of a tangent developable surface or a conical surface. One of the authors proved that tangential developable surfaces do not have generalized 1-type Gauss map and a conical surface of G-type can be constructed by the given functions f, g and the constant vector C ( [15]). Summing up our results, we obtain the following classification theorem.…”
Section: Classification Theoremmentioning
confidence: 99%
“…Regarding the Gauss map of finite-type, B.-Y. Chen and P. Piccini ( [6]) initiated to study submanifolds of Euclidean space with finite-type Gauss map and classified compact surfaces with 1-type Gauss map, that is, ∆G = λ(G + C), where C is a constant vector and λ ∈ R. Since then, quite a few works on ruled surfaces and ruled submanifolds with finite-type Gauss map in Euclidean space or pseudo-Euclidean space have been established ( [1,2,3,4,7,8,9,12,13,14,15]).…”
Section: Introductionmentioning
confidence: 99%
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