Geodesic
On local isometric immersions into complex and quaternionic projective spaces
Archivum mathematicum, Tome 47 (2011) no. 4, pp. 251-256 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We will prove that if an open subset of $\mathbb{C}{}P^{n}$ is isometrically immersed into $\mathbb{C}{}P^{m}$, with $m(4/3)n-2/3$, then the image is totally geodesic. We will also prove that if an open subset of $\mathbb{H}{}P^{n}$ isometrically immersed into $\mathbb{H}{}P^{m}$, with $m(4/3)n-5/6$, then the image is totally geodesic.
We will prove that if an open subset of $\mathbb{C}{}P^{n}$ is isometrically immersed into $\mathbb{C}{}P^{m}$, with $m(4/3)n-2/3$, then the image is totally geodesic. We will also prove that if an open subset of $\mathbb{H}{}P^{n}$ isometrically immersed into $\mathbb{H}{}P^{m}$, with $m(4/3)n-5/6$, then the image is totally geodesic.
Classification : 53C40
Keywords: submanifolds; homogeneous spaces; symmetric spaces 
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     title = {On local isometric immersions into complex and quaternionic projective spaces},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a1/}
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Rivertz, Hans Jakob. On local isometric immersions into complex and quaternionic projective spaces. Archivum mathematicum, Tome 47 (2011) no. 4, pp. 251-256. http://geodesic.mathdoc.fr/item/ARM_2011_47_4_a1/

[1] Agaoka, Y.: A note on local isometric imbeddings of complex projective spaces. J. Math. Kyoto Univ. 27 (3) (1987), 501–505. | MR | Zbl

[2] Agaoka, Y., Kaneda, E.: On local isometric immersions of Riemannian symmetric spaces. Tôhoku Math. J. 36 (1984), 107–140. | DOI | MR | Zbl

[3] Bourguignon, J., Karcher, H.: Curvature operators pinching estimates and geometric examples. Ann. Sci. École Norm. Sup. (4) 11 (1978), 71–92. | MR | Zbl

[4] Dajczer, M., Rodriguez, L.: On isometric immersions into complex space forms. VIII School on Differential Geometry (Portuguese) (Campinas, 1992), vol. 4, Mat. Contemp., 1993, pp. 95–98. | MR | Zbl

[5] Dajczer, M., Rodriguez, L.: On isometric immersions into complex space forms. Math. Ann. 299 (1994), 223–230. | DOI | MR | Zbl

[6] Gray, A.: A note on manifolds whose holonomy group is a subgroup of $Sp(n)\cdot Sp(1)$. Michigan Math. J. 16 (1969), 125–128. | MR

[7] Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, San Francisco and London, 1978, Ch. 4. | MR | Zbl

[8] Küpelî, D. N.: Notes on totally geodesic Hermitian subspaces of indefinite Kähler manifolds. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 43 (1) (1995), 1–7. | MR

[9] Rivertz, H. J.: On isometric and conformal immersions into Riemannian spaces. Ph.D. thesis, Department of Mathematics, University of Oslo, 1999.

[10] Tomter, P.: Isometric immersions into complex projective space. Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie, vol. 37, Adv. Stud. Pure Math., 2002, pp. 367–396. | MR | Zbl

[11] Wolf, J. A.: Correction to: The geometry and structure of isotropy irreducible homogeneous spaces. Acta Math. 152 (1984), 141–152. | DOI | MR | Zbl