A Moebius characterization of Veronese surfaces in S

LetMm be an umbilic-free submanifold in Sn with I and II as the first and second fundamental forms. An important Moebius invariant forMm in Moebius differential geometry is the so-calledMoebius formΦ, defined byΦ = −ρ−2 ∑i,α{Hα ,i +∑j (IIα ij −HαIij )ej (log ρ)}ωi⊗ eα , where {ei} is a local basis of the tangent bundle with dual basis {ωi}, {eα} is a local basis of the normal bundle, H = ∑α Heα is the mean curvature vector and ρ = √ m m−1‖II −HI‖. In this paper we prove that if x : S2 → Sn is an umbilics-free immersion of 2-sphere with vanishing Moebius form Φ, then there exists a Moebius transformation τ : Sn → Sn and a 2k-equator S2k ⊂ Sn with 2 ≤ k ≤ [n/2] such that τ ◦ x : S2 → S2k is the Veronese surface. Mathematics Subject Classification (2000): 53A30, 53C42, 53A10. 0. Introduction Let M be an umbilic-free submanifold in S with I and II as the first and second fundamental forms. Then the euclidean invariant Φ = −ρ−2 ∑ i,α {Hα ,i + ∑ j (I I α ij −HIij )ej (log ρ)}ωi ⊗ eα is also invariant under the Moebius transformation group in S, called Moebius form ofM, where {ei} is a local basis of the tangent bundle with dual basis {ωi}, {eα} is a local basis of the normal bundle, H = ∑α Heα is the mean curvature vector andρ = √ m m−1‖II−HI‖. ThisMoebius invariant plays an important role H. Li Department of Mathematical Sciences, Tsinghua Unviersity, Beijing 100084, People’s Republic of China (e-mail: [email protected]) C.P. Wang Department of Mathematics, Peking University, Beijing 1000871, People’s Republic of China (e-mail: [email protected]) F. Wu Department of Mathematics, Northern Jiaotong University, Beijing, 100044, People’s Republic of China (e-mail: [email protected]) The first author is supported by the project No.19701017 of NSFC and the second author is supported by 973 Project, RFDP, Quishi Award and DFG466-CHV-II3/127/0.