Totally real minimal 2-spheres in quaternionic projective space |
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Authors: | Yijun?He Email author" target="_blank">Changping?WangEmail author |
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Institution: | LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China |
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Abstract: | Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe
{I, J, K} of complex structures on ℍP
n
satisfying IJ = -JI = K,JK = -KJ = I,KI = -IK = J. A surface M ⊂ ℍP
n
is called totally real, if at each point p ∈ M the tangent plane T
p
M is perpendicular to I(T
p
M), J(T
p
M) and K(T
p
M). It is known that any surface M ⊂ ℝP
n
⊂ ℍP
n
is totally real, where ℝP
n
⊂ ℍP
n
is the standard embedding of real projective space in ℍP
n
induced by the inclusion ℝ in ℍ, and that there are totally real surfaces in ℍP
n
which don’t come from this way. In this paper we show that any totally real minimal 2-sphere in ℍP
n
is isometric to a full minimal 2-sphere in ℝP
2m
⊂ ℝP
n
⊂ ℍP
n
with 2m ≤ n. As a consequence we show that the Veronese sequences in ℝP
2m
(m ≥ 1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space. |
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Keywords: | quaternionic projective space totally real surfaces minimal surfaces |
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