The index of isolated umbilics on surfaces of non-positive curvature

Fontenele, Francisco; Xavier, Frederico

doi:10.4171/lem/61-1/2-6

Cited by 3 publications

(6 citation statements)

References 8 publications

(11 reference statements)

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“…Since λ = µ, the inequality (2.10) is equivalent to L E N E − M 2 E < 0 (that is, the Gaussian curvature of f λ is negative) on W \{o} for a sufficiently small neighborhood W of o. Thus, by the theorem in [3], the indices of the curvature line flows of f λ at o are non-positive. So, we obtain the conclusion.…”

Section: Sincementioning

confidence: 99%

“…where II f λ is the second fundamental form of the surface f λ induced by λ := µ in the Euclidean 3-space. Then, by applying the theorem in [3], we can conclude that the indices of the curvature line flows of f λ at o are non-positive, and so, the space-like surface g at o has the same property. By a straightforward computation, we have…”

Section: Using Theorem 23 We Prove Theorem a In The Introductionmentioning

confidence: 97%

“…Since the turn of the 21st century, Tari [4] proved an analogue of the Carathéodory conjecture for closed convex surfaces in L 3 , and Fontenele and Xavier [3] showed the non-positivity of the index of an isolated umbilic on a surface in E 3 which is negatively curved except at the umbilic. The present article, inspired by these works, investigates the behavior of the curvature line flows on space-like surfaces and time-like surfaces.…”

Section: Introductionmentioning

confidence: 99%

“…1 + m/2). Motivated by the main theorem of [3], we show the following: Proposition B. The index of an isolated umbilic o on a space-like surface in L 3 whose Gaussian curvature is positive except at o is non-positive.…”

Section: Introductionmentioning

confidence: 99%

“…By setting µ := λ, we define a regular space-like surface g in L 3 given by (2.3) with (2.2). By Theorem 2.3 (see also (2.7)), one of the two curvature line flows of g coincides with either of the eigen-flows of H λ , and the isolated scalar point of H λ corresponds to an isolated umbilic of g. So the index of the isolated umbilic of g is…”

Section: Using Theorem 23 We Prove Theorem a In The Introductionmentioning

confidence: 99%

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Umbilics of surfaces in the Minkowski 3-space

Tari¹

2013

J. Math. Soc. Japan

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In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz-Minkowski space L 3 . In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in L 3 is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer m there exists a germ of a space-like surface with an isolated C ∞ -umbilic (resp. C 1 -umbilic) of index (3 − m)/2 (resp. 1 + m/2). We also show that the indices of isolated umbilics of time-like surfaces in L 3 that are not the accumulation points of quasi-umbilics are always equal to zero. On the other hand, when quasiumbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.

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Section: Sincementioning

confidence: 99%

Section: Using Theorem 23 We Prove Theorem a In The Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Using Theorem 23 We Prove Theorem a In The Introductionmentioning

confidence: 99%

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Umbilics of surfaces in the Minkowski 3-space

Tari¹

2013

J. Math. Soc. Japan

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Umbilics of Surfaces in the Lorentz–Minkowski 3-Space

Ando,

Umehara

2023

Results Math

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In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz–Minkowski space $${{\mathbb {L}}}^3$$ L 3 . In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in $${{\mathbb {L}}}^3$$ L 3 is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer m, there exists a germ of a space-like surface with an isolated $$C^{\infty }$$ C ∞ -umbilic (resp. $$C^1$$ C 1 -umbilic) of index $$(3-m)/2$$ ( 3 - m ) / 2 (resp. $$1+m/2$$ 1 + m / 2 ). We also show that the indices of isolated umbilics of time-like surfaces in $${{\mathbb {L}}}^3$$ L 3 that are not the accumulation points of quasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.

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