Abstract
We introduce a quaternionic invariant for an inclusive immersion into a quaternionic manifold, which is a quaternionic object corresponding to the Willmore functional. The lower bound of this invariant is given by topological invariant and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternionic projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternionic invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point under any variations such that the induced complex structures do not vary.
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References
Alekseevsky, D., Marchiafava, S.: Quaternionic structures on a manifold and subordinated structures. Ann. Mat. Pura Appl. (4) 171, 205–273 (1996)
Alekseevsky, D., Marchiafava, S.: A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold. Ann. Mat. Pura Appl. (4) 184, 53–74 (2005)
Alekseevsky, D., Marchiafava, S., Pontecorvo, M.: Compatible complex structures on almost quaternionic manifolds. Trans. Am. Math. Soc. 351, 997–1014 (1999)
Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362, 425–461 (1978)
Besse, A.: Einstein Manifolds. Springer, Berlin (1987)
Burstall, F., Calderbank, D.: Conformal submanifold geometry I-III, arXiv:1006.5700
Burstall, F., Khemar, I.: Twistors, 4-symmetric spaces and integrable systems. Math Ann. 344, 451–461 (2009)
Bryant, R.L.: Conformal and minimal immersions of compact surfaces into 4-sphere. J. Differ. Geom. 17, 455–473 (1982)
Castro, I., Urbano, F.: Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes. Ann. Glob. Anal. Geom. 13, 59–67 (1995)
Friedrich, T.: On surfaces in four-spaces. Ann. Glob. Anal. Geom. 2, 275–287 (1984)
Friedrich, T.: The geometry of \(t\)-holomorphic surfaces in \(S^{4}\). Math. Nachr. 137, 49–62 (1988)
Fujimura, S.: \(Q\)-connections and their changes on an almost quaternion manifold. Hokkaido Math. J. 5, 239–248 (1976)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1994)
Hasegawa, K.: On surfaces whose twistor lifts are harmonic sections. J. Geom. Phys. 57, 1549–1566 (2007)
Hasegawa, K.: Twistor holomorphic affine surfaces and projective invariants. SUT J. Math. 50, 325–341 (2014)
Kobak, P., Loo, B.: Moduli of quaternionic superminimal immersions of 2-spheres into quaternionic projective spaces. Ann. Glob. Anal. Geom. 16, 527–541 (1998)
Salamon, S.: Harmonic and Holomorphic Maps. Geometry Seminar “Luigi Bianchi” II-1984, 161–224, Lecture Notes in Mathematics, 1164. Springer, Berlin (1985)
Tsukada, K.: Transversally complex submanifolds of a quaternion projective space, preprint
Wood, C.M.: Harmonic sections of homogeneous fibre bundles. Differ. Geom. Appl. 19, 193–210 (2003)
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Dedicated to Professor Naoto Abe on the occasion of his retirement
This work is partially supported by JSPS KAKENHI Grant Number 15K04839.
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Hasegawa, K. An inclusive immersion into a quaternionic manifold and its invariants. manuscripta math. 154, 527–549 (2017). https://doi.org/10.1007/s00229-017-0928-5
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DOI: https://doi.org/10.1007/s00229-017-0928-5