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Minimal Legendrian Surfaces in the Five-Dimensional Heisenberg Group Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts

Authors : L. F. Martins, K. Saji, M. Umehara, K. Yamada

Published in: Geometry and Topology of Manifolds

Publisher: Springer Japan

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Abstract

We define cuspidal curvature \(\kappa _c\) (resp. normalized cuspidal curvature \(\mu _c\)) along cuspidal edges (resp. at a swallowtail singularity) in Riemannian 3-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product \(\kappa _\varPi ^{}\) called the product curvature (resp. \(\mu _\varPi ^{}\) called normalized product curvature) of \(\kappa _c\) (resp. \(\mu _c\)) and the limiting normal curvature \(\kappa _\nu \) is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider the limiting behavior of \(\kappa _\varPi ^{}\) when cuspidal edges accumulate to other singularities. Moreover, several new geometric invariants of cuspidal edges and swallowtails are given.

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previous chapter The Yau-Tian-Donaldson Conjecture for General Polarizations, I
next chapter The Calabi Invariant and Central Extensions of Diffeomorphism Groups
Appendix
Available only for authorised users
Footnotes
1
These invariants can be treated as invariants of a certain class of positive semi-definite metrics, see [7, 14, 15].
 
2
This might be considered as a variant of Gauss’ Theorema Egregium.
 
3
This example was suggested by the referee.
 
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Metadata
Title
Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts
Authors
L. F. Martins
K. Saji
M. Umehara
K. Yamada
Copyright Year
2016
Publisher
Springer Japan
Book
Geometry and Topology of Manifolds

Print ISBN: 978-4-431-56019-7

Electronic ISBN: 978-4-431-56021-0

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-4-431-56021-0_14

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