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2021 Relating second order geometry of manifolds through projections and normal sections
P. Benedini Riul, R. Oset Sinha
Publ. Mat. 65(1): 389-407 (2021). DOI: 10.5565/PUBLMAT6512114

Abstract

We use normal sections to relate the curvature locus of regular (resp. singular corank $1$) $3$-manifolds in $\mathbb{R}^6$ (resp.\ $\mathbb R^5$) with regular (resp.\ singular corank $1$) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example, we show how to generate a Roman surface by a family of ellipses different to Steiner's way. We also study the relations between the regular and singular cases through projections. We show that there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular, we define asymptotic directions for singular corank $1$ $3$-manifolds in $\mathbb R^5$ and relate them to asymptotic directions of regular $3$-manifolds in $\mathbb R^6$ and singular corank $1$ surfaces in $\mathbb R^4$.

Funding Statement

P. Benedini Riul was supported by FAPESP Grant 2019/00194-6. R.Oset Sinha was partially supported by MICINN Grant PGC2018-094889-B-I00 and GVA Grant AICO/2019/024.

Citation

Download Citation

P. Benedini Riul. R. Oset Sinha. "Relating second order geometry of manifolds through projections and normal sections." Publ. Mat. 65 (1) 389 - 407, 2021. https://doi.org/10.5565/PUBLMAT6512114

Information

Received: 4 December 2019; Revised: 8 July 2020; Published: 2021
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185839
Digital Object Identifier: 10.5565/PUBLMAT6512114

Subjects:
Primary: 57R45
Secondary: 53A05 , 58K05

Keywords: curvature locus , immersed $3$-manifolds , immersed surfaces , normal sections , projections , singular corank $1$ manifolds

Rights: Copyright © 2021 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.65 • No. 1 • 2021
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