Einige Bemerkungen zur Geometrie transnormaler Mannigfaltigkeiten

Wegner, Bernd

doi:10.4310/jdg/1214435991

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…This concept is studied in the papers [7], [8], [10], [11], [9]. Results of [7], [8], [10], [11] show that any transnormal immersion of a compact manifold must be an embedding or there exists a covering map π :…”

Section: Transnormal Embeddings With Flat Normal Bundlementioning

confidence: 99%

“…This set is called the generating frame at p. It is proved in [11] that there is a positive integer r such that Φ(p) consists of r elements ∀p ∈ M . This holds whether M is compact or not, but complete.…”

Section: Transnormal Embeddings With Flat Normal Bundlementioning

confidence: 99%

“…It is shown in [11] that the subbundle of the normal bundle in which the fibres are spanned by the generating frames is locally flat. We can deduce that if the generating frame at a point spans (k − 1)-subplane of the normal plane at that point (in which case r ≥ k) then it holds for all the points of the manifold and in such a case any transnormal embedding has locally flat normal bundle.…”

Section: Remark 23mentioning

confidence: 99%

“…For p ∈ M , let Φ 0 (p) be the set of minima of L f (p) . It is shown in [11] that ∀p, q ∈ M, Φ 0 (p) is isometric to Φ 0 (q). Therefore ∀p ∈ M, L f (p) has a unique minimum.…”

Section: So We Havementioning

confidence: 99%

“…Then, define ξ(p) = f (p ) − f (p), this is a normal direction on ν p (f ) such that f (p) + ξ(p) = f (p ) ∈ Φ(p). Now, as in Remark 2.2, extend this direction to a parallel normal field ξ on M. This will give us a parallel normal field ξ such that f ξ (q) ∈ Φ(q), ∀q ∈ M, [11]. Since Φ(q) ∩ F q = ∅, ∀q ∈ M, it follows that f ξ is an immersion and f ξ has index λ.…”

Section: So We Havementioning

confidence: 99%

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The push-out space of transnormal immersions

Kaya

2010

Arch. Math.

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Here we study the push-out space of a transnormal immersion of an m-dimensional compact manifold M without boundary into the Euclidean (m + k)-space and show that for an r-transnormal immersion with flat normal bundle, the push-out space has at most r components and this number can be attained in some cases, and any two components are isometric to each other. Mathematics Subject Classification (2000). Primary 53C40; Secondary 53C42.

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Section: Transnormal Embeddings With Flat Normal Bundlementioning

confidence: 99%

Section: Transnormal Embeddings With Flat Normal Bundlementioning

confidence: 99%

Section: Remark 23mentioning

confidence: 99%

“…For p ∈ M , let Φ 0 (p) be the set of minima of L f (p) . It is shown in [11] that ∀p, q ∈ M, Φ 0 (p) is isometric to Φ 0 (q). Therefore ∀p ∈ M, L f (p) has a unique minimum.…”

Section: So We Havementioning

confidence: 99%

Section: So We Havementioning

confidence: 99%

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The push-out space of transnormal immersions

Kaya

2010

Arch. Math.

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Self-parallel and transnormal curves

Wegner

1991

Geom Dedicata

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The construction of examples of self-parallel curves from a closed central curve given by F. J. Craveiro de Carvalho and S. A. Robertson is extended to general dosed regular curves without the assumption of non-vanishing curvature made there. Furthermore, every self-parallel curve in 3-space is shown to be of the type obtained by this construction, if the order of the selfparallel group is greater than 2. These considerations are used to present examples for transnormal curves in 4-space with arbitrarily high degree of transnormality, disproving a longstanding conjecture of M. C. Irwin.The notion of parallel immersions into Euclidean space has been introduced by Farran and Robertson [3]. In particular, self-parallel immersions are generalizations of the so-called transnormal submanifolds which have been studied in several papers (see the survey [5]). In the latter case very detailed results have been obtained for closed curves.The aim of this note is to study parallelism and self-parallelism for curves in Euclidean space, complementing results obtained by Craveiro de Carvalho and Robertson [1]. The more geometric interpretation of the parallelism of immersions introduced in [7] or I-8] is shown to give a better insight into the structure of self-parallel curves. The construction of examples given in [1] will be extended to the general case without the restriction on the curvature needed there. Furthermore, every self-parallel curve can be obtained in this way from a suitable central curve. Relations between the geometrical data of this central curve and those of the original curve will be obtained.Returning to the case of transnormal curves, these considerations yield a new possibility for the construction of transnormal curves. This is used to show that there are transnormal curves in E 4 of an arbitrarily high degree of transnormality, solving an old problem posed by Irwin [4]. Also injective infinitely transnormal immersions of the real line into E 4 are possible, while this can be excluded in E 3. PRELIMINARIESLet c: D ~ E" be a regular parametrized curve of sufficiently high differentiability in Euclidean n-space, D = R or S 1. As usual let s denote the arc length of e, T(s) its unit tangent field, ~ its (first) curvature and v(t) = N(t) + e(t) the affine normal space of c at t, where N(t) denotes the normal vector space of c at t.

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Projections of transnormal manifolds

Wegner

1989

J Geom

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Einige Bemerkungen zur Geometrie transnormaler Mannigfaltigkeiten

Cited by 7 publications

References 9 publications

The push-out space of transnormal immersions

The push-out space of transnormal immersions

Self-parallel and transnormal curves

Projections of transnormal manifolds

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