VARIATIONAL PROBLEMS FOR INTEGRAL INVARIANTS OF THE SECOND FUNDAMENTAL FORM OF A MAP BETWEEN PSEUDO-RIEMANNIAN MANIFOLDS

[1] C.B. Allendoerfer and A. Weil: The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math.

Soc. 53 (1943), 101–129.

[2] R.L. Bryant: Minimal surfaces of constant curvature in Sn , Trans. Amer. Math. Soc. 290 (1985), 259–271.

[3] T.E. Cecil and P.J. Ryan: Geometry of hypersurfaces, Springer Monographs in Mathematics, Springer, New

York, 2015.

[4] B.-Y. Chen: An invariant of conformal mappings, Proc. Amer. Math. Soc. 40 (1973), 563–564.

[5] B.-Y. Chen: Some conformal invariants of submanifolds and their applications, Boll. Unione Mat. Ital. (4)

10 (1974), 380–385.

[6] B.-Y. Chen: Pseudo-Riemannian geometry, δ-invariants and applications, World Scientific, Hackensack,

NJ, 2011.

[7] B.-Y. Chen: Recent developments in δ-Casorati curvature invariants, Turkish J. Math. 45 (2021), 1–46.

[8] J. Eells and L. Lemaire: Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics 50, Amer. Math. Soc., Providence, RI, 1983.

[9] R. Howard: The kinematic formula in Riemannian homogeneous spaces, Mem. Amer. Math. Soc. 106

(1993), no. 509.

[10] T. Ichiyama, J. Inoguchi and H. Urakawa: Bi-harmonic maps and bi-Yang-Mills fields, Note Mat. 28 (2009),

[2008 on verso], suppl. 1, 233–275.

[11] G. Jiang: 2-harmonic maps and their first and second variational formulas, Translated from the Chinese

by Hajime Urakawa, Note Mat. 28 (2009), [2008 on verso], suppl. 1, 209–232.

[12] H.J. Kang, T. Sakai and Y.J. Suh: Kinematic formulas for integral invariants of degree two in real space

forms, Indiana Univ. Math. J. 54 (2005), 1499–1519.

[13] Y. Kitagawa: Periodicity of the asymptotic curves on flat tori in S3 , J. Math. Soc. Japan 40 (1988), 457–476.

[14] Y. Kitagawa: Isometric deformations of flat tori in the 3-sphere with nonconstant mean curvature, Tohoku

Math. J. (2) 52 (2000), 283–298.

[15] K. Kenmotsu: On minimal immersion of R2 into SN , J. Math. Soc. Japan 28 (1976), 182–191.

[16] R. Moser: A variational problem pertaining to biharmonic maps, Comm. Partial Differential Equations 33

(2008), 1654–1689.

[17] H. Weyl: On the volume of tubes, Amer. J. Math. 61 (1939), 461–472.

Variational Problems for Integral Invariants

901

Rika Akiyama

Department of Mathematical Sciences

Tokyo Metropolitan University

Minami-Osawa 1–1, Hachioji

Tokyo, 192–0397

Japan

e-mail: akiyama-rika@ed.tmu.ac.jp

Takashi Sakai

Department of Mathematical Sciences

Tokyo Metropolitan University

Minami-Osawa 1–1, Hachioji

Tokyo, 192–0397

Japan

e-mail: sakai-t@tmu.ac.jp

Yuichiro Sato

Academic Support Center

Kogakuin University

Nakano-cho, 2665–1, Hachioji

Tokyo, 192–0015

Japan

e-mail: kt13699@ns.kogakuin.ac.jp

yuichiro-sato@tmu.ac.jp

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