论文标题
变形磁性标记的长度光谱刚度
Deformative Magnetic Marked Length Spectrum Rigidity
论文作者
论文摘要
令$ m $为封闭的表面，让$ \ {g_s \ | \ s \ in（-ε，ε）\} $是$ m $的平滑单参数riemannian指标家族。也让$ \ {κ_s：m \ rightarrow \ mathbb {r} \ | \ s \ in（-ε，ε）\} $是$ m $上的平滑单参数功能家族。然后家庭$ \ {（g_s，κ_s）\ | \ s \ in（-ε，ε）\} $在$ tm $上产生了磁流的家族。我们表明，如果（-ε，ε）$ $ s \的磁性曲率为负，并且每个周期轨道的长度随着参数$ s $的变化而保持恒定，那么存在一个光滑的差异属性$ \ \ {f_s：m {m \ rightarrow m \ rightarrow m \ | | \ s \ in（-ε，ε）\} $，使得$ f_s^*（g_s）= g_0 $和$ f_s^*（κ_s）=κ__0$。这将Guillemin和Kazhdan的结果推广到磁流的设置。
Let $M$ be a closed surface and let $\{g_s \ | \ s \in (-ε, ε)\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{κ_s : M \rightarrow \mathbb{R} \ | \ s \in (-ε, ε)\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s, κ_s) \ | \ s \in (-ε, ε)\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s \in (-ε, ε)$ and the lengths of each periodic orbit remains constant as the parameter $s$ varies, then there exists a smooth family of diffeomorphisms $\{f_s : M \rightarrow M \ | \ s \in (-ε, ε)\}$ such that $f_s^*(g_s) = g_0$ and $f_s^*(κ_s) = κ_0$. This generalizes a result of Guillemin and Kazhdan to the setting of magnetic flows.
