Papers
Topics
Authors
Recent
Search
2000 character limit reached

Half-dimensional immersions into the para-complex projective space and Ruh-Vilms type theorems

Published 20 May 2024 in math.DG | (2405.11771v1)

Abstract: In this paper we study isometric immersions $f:Mn \to {\mathbb {C}{\prime}}!Pn$ of an $n$-dimensional pseudo-Riemannian manifold $Mn$ into the $n$-dimensional para-complex projective space ${\mathbb {C}{\prime}}!Pn$. We study the immersion $f$ by means of a lift $\mathfrak f$ of $f$ into a quadric hypersurface in ${S{2n+1}_{n+1}}$. We find the frame equations and compatibility conditions. We specialize these results to dimension $n = 2$ and a definite metric on $M2$ in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (24)
  1. Homogeneous para-Kählerian Einstein manifolds. Russian Math. Surveys 64(1):1–43, 2009.
  2. Henri Anciaux, Maikel Antonio Samuays. Lagrangian submanifolds in para-complex Euclidean space. Bull. Belg. Math. Soc. Simon Stevin 23(3):421–437, 2016.
  3. Malcolm Black. Harmonic maps into homogeneous spaces. Pitman Research Notes in Mathematics Series, 255. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
  4. Shaoping Chang. On Hamiltonian stable minimal Lagrangian surfaces in ℂ⁢P2ℂsuperscript𝑃2\mathbb{C}P^{2}blackboard_C italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. J. Geom. Anal. 10:243–255, 2000.
  5. A survey on paracomplex geometry. Rocky Mountain J. Math. 26:83–115, 1996.
  6. Josef F. Dorfmeister, Shimpei Kobayashi. Timelike minimal Lagrangian surfaces in the indefinite complex hyperbolic two-space. Ann. Mat. Pura Appl.(4) 200(2):521–546, 2021.
  7. Survey on real forms of the complex A(2)2-Toda equation and surface theory. Complex Manifolds 6(1):194–227, 2019.
  8. Ruh-Vilms theorems for minimal surfaces without complex points and minimal Lagrangian surfaces in ℂ⁢P2ℂsuperscript𝑃2\mathbb{C}P^{2}blackboard_C italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Math. Z. 296(3-4):1751–1775, 2020.
  9. Josef F. Dorfmeister, Hui Ma. Explicit expressions for the Iwasawa factors, the metric and the monodromy matrices for minimal Lagrangian surfaces in ℂ⁢ℙ2ℂsuperscriptℙ2\mathbb{CP}^{2}blackboard_C blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. In Dynamical Systems, Number Theory and Applications, chapter 2, pages 19–47. World Scientific, 2016.
  10. Josef F. Dorfmeister, Hui Ma. A new look at equivariant minimal Lagrangian surfaces in ℂ⁢ℙ2ℂsuperscriptℙ2\mathbb{CP}^{2}blackboard_C blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. In Akito Futaki, Reiko Miyaoka, Zizhou Tang, and Weiping Zhang, editors, Geometry and Topology of Manifolds: 10th China-Japan Conference 2014, volume 154 of Springer Proceedings in Mathematics and Statistics, pages 97–126. Springer, Tokyo, 2016.
  11. Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6(4):633–668, 1998.
  12. The geometry of a bi-Lagrangian manifold. Differ. Geom. Appl., 24(1):33–59, 2006.
  13. Otto Forster. Lectures on Riemann surfaces. Graduate Texts in Mathematics, 81. Springer-Verlag, New York, 1991.
  14. Pedro Martinez Gadea, A. Montesinos Amilibia. Spaces of constant para-holomorphic sectional curvature. Pacific J. Math., 136(1):85–101, 1989.
  15. Pedro Martinez Gadea, A. Montesinos Amilibia. The paracomplex projective spaces as symmetric and natural spaces. Indian J. Pure Appl. Math., 23(4):261–275, 1992.
  16. Roland Hildebrand. The cross-ratio manifold: a model of centro-affine geometry. Int. Electron. J. Geom., 4(2):32–62, 2011.
  17. Roland Hildebrand. Half-dimensional immersions in para-Kähler manifolds. Int. Electron. J. Geom., 4(2):85–113, 2011.
  18. Soji Kaneyuki, Masato Kozai. Paracomplex structures and affine symmetric spaces. Tokyo J. Math. 8(1):81–98, 1985.
  19. Hui Ma, Yujie Ma. Totally real minimal tori in ℂ⁢ℙ2ℂsuperscriptℙ2\mathbb{CP}^{2}blackboard_C blackboard_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Math. Z. 249(2):241–267, 2005.
  20. Ian McIntosh. Special Lagrangian cones in ℂ3superscriptℂ3\mathbb{C}^{3}blackboard_C start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and primitive harmonic maps. J. London Math. Soc. 67(2):769–789, 2003.
  21. Andrei E. Mironov. Finite-gap minimal Lagrangian surfaces in ℂ⁢P2ℂsuperscript𝑃2\mathbb{C}P^{2}blackboard_C italic_P start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, volume 3 of OCAMI Studies Series, pp. 185–196, 2010.
  22. Hans Samelson. Orientability of hypersurfaces in ℝnsuperscriptℝ𝑛\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Proc. Amer. Math. Soc. 22:301–302, 1969.
  23. Katsumi Nomizu, Takeshi Sasaki. Affine Differential Geometry: Geometry of Affine Immersions, volume 111 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1994.
  24. Hermann Weyl. Zur Infinitesimalgeometrie: Einordnung der projektiven und konformen Auffassung. Göttinger Nachr., pages 99–112, 1921.

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 2 tweets with 0 likes about this paper.

Sign Up for Free