Horospherical平均曲率函数与D'Atri空间
Horospherical mean curvature functions and D'Atri spaces
AI总结：
本文研究满足特定horospherical平均曲率函数性质的无共轭点单连通黎曼流形，证明附加温和曲率条件下秩一流形渐近调和、紧秩一情形为负曲率局部对称空间，且同条件下秩一D'Atri空间调和，推广了Druetta的相关结果。
摘要：
我们研究了无共轭点的单连通黎曼流形，其horospherical平均曲率函数满足连续性、可逆性且在测地流下不变。我们证明，在温和的附加曲率张量条件下，该族中的秩一流形自动是渐近调和的。特别地，此类紧秩一流形必为负曲率的局部对称空间。此外，我们还证明在相同条件下，无共轭点的秩一D'Atri空间是调和的。此前Druetta曾针对某些齐次D'Atri空间证明过此类结果。
英文摘要：
We consider simply connected Riemannian manifolds without conjugate points for which the horospherical mean curvature function is continuous, reversible and invariant under the geodesic flow. We show under mild additional curvature tensor conditions that rank one manifolds in this family are automatically asymptotically harmonic. In particular, compact rank one manifolds of this kind must be locally symmetric spaces of negative curvature. Moreover, we show under the same conditions that rank one D'Atri spaces without conjugate points are harmonic. An earlier result of this type was proved by Druetta for certain homogeneous D'Atri spaces.
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