A Mobius characterization of submanifolds

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In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere <i>S</i><sup><i>n</i>+<i>p</i></sup>(1). First of all, we proved that, for an <i>n</i>-dimensional (<i>n</i>≥2) submanifold <b>x</b>:<i>M</i>$¥mapsto$<i>S</i><sup><i>n</i>+<i>p</i></sup>(1) without umbilical points and with vanishing Möbius form Φ, if (<i>n</i>-2)||<b><i>Ã</i></b>||≤$¥sqrt{¥smash[b]{¥frac{n-1}{n}}} ¥big¥{ nR-¥frac{1}{n}[(n-1)¥big( 2-¥frac{1}{p} ¥big)-1] ¥big¥}$ is satisfied, then, <b>x</b> is Möbius equivalent to an open part of either the Riemannian product <i>S</i><sup><i>n</i>-1</sup>(<i>r</i>)×<i>S</i><sup>1</sup>($¥sqrt{1-r^2}$) in <i>S</i><sup><i>n</i>+1</sup>(1), or the image of the conformal diffeomorphism σ of the standard cylinder <i>S</i><sup><i>n</i>-1</sup>(1)×<b><i>R</i></b> in <b><i>R</i></b><sup><i>n</i>+1</sup>, or the image of the conformal diffeomorphism τ of the Riemannian product <i>S</i><sup><i>n</i>-1</sup>(<i>r</i>)×<b><i>H</i></b><sup>1</sup>($¥sqrt{1+r^2}$) in <b><i>H</i></b><sup><i>n</i>+1</sup>, or <b>x</b> is locally Möbius equivalent to the Veronese surface in <i>S</i><sup>4</sup>(1). When <i>p</i>=1, our pinching condition is the same as in Main Theorem of Hu and Li [<b>6</b>], in which they assumed that <i>M</i> is compact and the Möbius scalar curvature <i>n</i>(<i>n</i>-1)<i>R</i> is constant. Secondly, we consider the Möbius sectional curvature of the immersion <b>x</b>. We obtained that, for an <i>n</i>-dimensional compact submanifold <b>x</b>:<i>M</i>$¥mapsto$<i>S</i><sup><i>n</i>+<i>p</i></sup>(1) without umbilical points and with vanishing form Φ, if the Möbius scalar curvature <i>n</i>(<i>n</i>-1)<i>R</i> of the immersion <b>x</b> is constant and the Möbius sectional curvature <i>K</i> of the immersion <b>x</b> satisfies <i>K</i>≥0 when <i>p</i>=1 and <i>K</i>>0 when <i>p</i>>1. Then, <b>x</b> is Möbius equivalent to either the Riemannian product <i>S</i><sup><i>k</i></sup>(<i>r</i>)×<i>S</i><sup><i>n</i>-<i>k</i></sup>($¥sqrt{1-r^2}$), for <i>k</i>=1, 2, …, <i>n</i>-1, in <i>S</i><sup><i>n</i>+1</sup>(1); or <b>x</b> is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in <i>S</i><sup><i>n</i>+<i>p</i></sup>(1).

収録刊行物

  • Journal of the Mathematical Society of Japan  

    Journal of the Mathematical Society of Japan 58(3), 903-925, 2006-07-01 

    The Mathematical Society of Japan

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各種コード

  • NII論文ID(NAID)
    10018381194
  • NII書誌ID(NCID)
    AA0070177X
  • 本文言語コード
    ENG
  • 資料種別
    ART
  • ISSN
    00255645
  • NDL 記事登録ID
    7987208
  • NDL 雑誌分類
    ZM31(科学技術--数学)
  • NDL 請求記号
    Z53-A209
  • データ提供元
    CJP書誌  NDL  J-STAGE 
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