Criteria for singularities of smooth maps from the plane into the plane and their applications
Kentaro Saji
Source: Hiroshima Math. J. Volume 40, Number 2
(2010), 229-239.
Abstract
We give useful criteria for lips, beaks and swallowtail singularities of a smooth map from the plane into the plane. As an application of the criteria, we discuss the singularities for a Cauchy problem of single conservation law.
First Page:
Show
Hide
Primary Subjects:
57R45
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.hmj/1280754423
Mathematical Reviews number (MathSciNet): MR2680658
Zentralblatt MATH identifier: 05831951
References
S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces, Math. Z. 259 (2008), 827--848.
Mathematical Reviews (MathSciNet): MR2403743
Digital Object Identifier: doi:10.1007/s00209-007-0250-0
T. Gaffney, The structure of $T\mathcal A(f)$, classification and an application to differential geometry, In singularities, Part I, Proc. Sympos. in Pure Math. 40 (1983), Amer. Math. Soc., 409--427.
Mathematical Reviews (MathSciNet): MR713081
Zentralblatt MATH: 0523.58008
G. Ishikawa and Y. Machida, Singularities of improper affine spheres and surfaces of constant Gaussian curvature, Int. J. Math. 17 (2006), 269--293.
Mathematical Reviews (MathSciNet): MR2215151
Zentralblatt MATH: 1093.53067
Digital Object Identifier: doi:10.1142/S0129167X06003485
S. Izumiya and G. T. Kossioris, Geometric singularities for solutions of single conservation laws, Arch. Rational Mech. Anal. 139 (1997), 255--290.
Mathematical Reviews (MathSciNet): MR1480242
Zentralblatt MATH: 0907.35082
Digital Object Identifier: doi:10.1007/s002050050053
S. Izumiya, K. Saji and M. Takahashi, Horospherical flat surfaces in Hyperbolic $3$-space, to appear in J. Math. Soc. Japan.
S. Izumiya, K. Saji and N. Takeuchi, Circular surfaces, Advances in Geometry 7 (2007), 295--313.
Mathematical Reviews (MathSciNet): MR2314822
Zentralblatt MATH: 1123.53005
Digital Object Identifier: doi:10.1515/ADVGEOM.2007.017
M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic $3$-space, Pacific J. Math. 221 (2005), no. 2, 303--351.
Mathematical Reviews (MathSciNet): MR2196639
Zentralblatt MATH: 1110.53044
Digital Object Identifier: doi:10.2140/pjm.2005.221.303
M. Kokubu, W. Rossman, M. Umehara and K. Yamada, Flat fronts in hyperbolic 3-space and their caustics, J. Math. Soc. Japan 50 (2007), 265--299.
Mathematical Reviews (MathSciNet): MR2302672
Zentralblatt MATH: 1120.53036
Digital Object Identifier: doi:10.2969/jmsj/1180135510
Project Euclid: euclid.jmsj/1180135510
J. H. Rieger, Families of maps from the plane to the plane, J. London Math. Soc. (2) (1987) 351--369.
Mathematical Reviews (MathSciNet): MR906153
Zentralblatt MATH: 0639.58007
Digital Object Identifier: doi:10.1112/jlms/s2-36.2.351
K. Saji, T. Sasaki and M. Yoshida, Hyperbolic Schwarz map of the confluent hypergeometric differential equation, J. Math. Soc. Japan 61, No. 2, (2009), 559--578.
Mathematical Reviews (MathSciNet): MR2532901
Zentralblatt MATH: 05573650
Digital Object Identifier: doi:10.2969/jmsj/06120559
Project Euclid: euclid.jmsj/1242220722
K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. of Math. 169 (2009), 491--529.
Mathematical Reviews (MathSciNet): MR2480610
Zentralblatt MATH: 1177.53014
Digital Object Identifier: doi:10.4007/annals.2009.169.491
K. Saji, M. Umehara and K. Yamada, $A_k$ singularities of wave fronts, Math. Proc. Camb. Phils. Soc. 146, (3), (2009), 731--746.
Mathematical Reviews (MathSciNet): MR2496355
Digital Object Identifier: doi:10.1017/S0305004108001977
H. Whitney, On singularities of mappings of Euclidean spaces I. Mappings of the plane into the plane, Ann. of Math. 62, (1955), 374--410.
Mathematical Reviews (MathSciNet): MR73980
Digital Object Identifier: doi:10.2307/1970070
JSTOR: links.jstor.org
Hiroshima Mathematical Journal
