## Rocky Mountain Journal of Mathematics

### A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank

#### Abstract

We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorphism by using the Hamilton vector fields.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 1 (2019), 335-354.

Dates
First available in Project Euclid: 10 March 2019

https://projecteuclid.org/euclid.rmjm/1552186964

Digital Object Identifier
doi:10.1216/RMJ-2019-49-1-335

Mathematical Reviews number (MathSciNet)
MR3921879

Zentralblatt MATH identifier
07036630

#### Citation

Saji, Kentaro; Tsuchida, Asahi. A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank. Rocky Mountain J. Math. 49 (2019), no. 1, 335--354. doi:10.1216/RMJ-2019-49-1-335. https://projecteuclid.org/euclid.rmjm/1552186964

#### References

• V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of differentiable maps, $1$, Mono. Math. 82 (1985).
• N. Dutertre and T. Fukui, On the topology of stable maps, J. Math. Soc. Japan 66 (2014), 161–203.
• A. Honda, Isometric immersions with singularities between space forms of the same positive curvature, J. Geom. Anal. 27 (2017), 2400–2417.
• M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic space, Pacific J. Math. 221 (2005), 303–351.
• H.I. Levine, The singularities, $S_1^q$, Illinois J. Math. 8 (1964), 152–168.
• K. Saji, Criteria for singularities of smooth maps from the plane into the plane and their applications, Hiroshima Math. J. 40 (2010), 229–239.
• ––––, Criteria for Morin singularities for maps into lower dimensions, and applications, Contemp. Math. 675 (2016), 315–336.
• K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. Math. 169 (2009), 491–529.
• ––––, $A_k$ singularities of wave fronts, Math. Proc. Cambr. Philos. Soc. 146 (2009), 731–746.
• ––––, Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts, J. Geom. Anal. 22 (2012), 383–409.
• ––––, An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications, J. Math. Soc. Japan 69 (2017), 417–457.
• T. Schmah and C. Stoica, Saari's conjecture is true for generic vector fields, Trans. Amer. Math. Soc. 359 (2007), 4429–4448.
• H. Whitney, On singularities of mappings of Euclidean spaces, I, Mappings of the plane into the plane, Ann. Math. 62 (1955), 374–410.