Abstract
In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by \(-c^2\) is of Anosov type, then the constant of contraction of the flow is \(\ge e^{-c}\). Moreover, if M has a finite volume, the equality holds if and only if the sectional curvature is constant. We also apply this result to get a certain rigidity for bi-Lipschitz, and consequently, for \(C^1\)-conjugacy between two geodesic flows.
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Notes
We can think in \(\alpha \)- equivalence (\(\alpha \)-conjugacy), when there are two constants \(C_1\) and \(C_2\) such that \(C_1\cdot d(x,y)^{\alpha }\le D(f(x),f(y))\le C_2\cdot d(x,y)^{\alpha }\), but such definition do not make any sense for \(C^1\) flows.
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Acknowledgements
The authors would like to thank François Ledrappier for his useful comments during the preparation of this work and the anonymous referees for the great suggestions that improved the paper. Ítalo Melo was partially supported by FAPEPI (Brazil) and CNPq (Brazil) and Sergio Romaña was supported by CNPq and Faperj - Bolsa Jovem Cientista do Nosso Estado No. E-26/201.432/2022.
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Í. Melo Partially supported by CAPES-BR. S. Romaña CNPq and Bolsa Jovem Cientista do Nosso Estado No. E-26/201.432/2022.
Appendix A: Comparing Distances Between a Manifold and Its Submanifolds
Appendix A: Comparing Distances Between a Manifold and Its Submanifolds
This appendix is devoted to prove the Lemma A.1. This a general lemma that can be in other context.
Lemma A.1
Let Q be a complete Riemannian manifold and P a compact submanifold of Q. Consider d the natural distance in Q and \(d_P\) the extrinsic distance in P. Then there is \(\delta >0\) such that for each \(x\in P\)
where \(B^{P}_{\delta }(x)\) is the ball of center x and radius \(\epsilon \) in P.
Proof
Since P is a compact submanifold of Q, then the injectivity radius \(r_{P}\) of P with extrinsic metric is a positive number. Denote by SP the unitary bundle of P and consider the non-negative real function \({\mathcal {H}}:SP\times [0,r_p] \rightarrow {\mathbb {R}}\) defined by
where \(\text {exp}^P_{x}\) denotes the exponential map of P.
It is easy to see that \({\mathcal {H}}\) is continuous for all ((x, v), t) with \(t\ne 0\). We state that \({\mathcal {H}}\) is continuous at any point ((x, v), 0). To prove that, we used the compactness of P and the following claim:
Claim: For each \(x\in P\) the function \({\mathcal {H}}_{x}:T^{1}_{x}P\times [0,r_P] \rightarrow {\mathbb {R}}\) defined by \({\mathcal {H}}_{x}(v,t):={\mathcal {H}}((x,v),t)\) is uniformly continuous function in \(T^{1}_{x}P\times [0,r_P]\), where \(T^{1}_{x}P\) is the set of unitary tangent vectors of P at x.
Proof of Claim
By definition of \({\mathcal {H}}_{x}\) and compactness of \(T^{1}_{x}P\times [0,r_P]\), we only need to prove that \(\displaystyle \lim \nolimits _{n\rightarrow +\infty }{\mathcal {H}}_{x}(v_n,t_n)={\mathcal {H}}_{x}(v,0)=1\), for any sequence \((v_n, t_n)\in T^{1}_{x}P\times [0,r_P]\), which converges to (v, 0). In fact: first note that if \((x,v)\in SP\) and \(\alpha (t)=\text {exp}^{P}_{x}\,tv\), then from Lemma 4.1 we have that
since \(||\alpha '(0)||_{P}\) is the norm with restrict metric of \(P\subset Q\) which is equal to \(||\alpha '(0)||\).
As \({\mathcal {H}}_{x}(v_n,0)=1\) then we can assume, without loss of generality, that \(t_n\ne 0\) for all n.
For n large enough, consider the family of \(C^1\)-curves \(\gamma _{n}(t) = \text {exp}^{P}_{x} (t_nv + t t_n(v_n - v))\), \(t\in [0,1]\), and note that
where \(K:=\displaystyle \sup \nolimits _{\{v\in T_{x}P: ||v||\le 2r_P\}} || D (\text {exp}^{P}_{x})_{v}||\). Hence, since \(\displaystyle \lim \nolimits _{n \rightarrow +\infty }{\mathcal {H}}_{x}(v,t_n)=1\), then
In particular, we have
Therefore \(\displaystyle \lim \nolimits _{ n \rightarrow + \infty }{\mathcal {H}}_{x}(v_n,t_n) = 1\) as desired. \(\square \)
The last claim and compactness of P allow us to conclude the continuity of \({\mathcal {H}}\) in every point ((x, v), 0) and consequently the uniformly continuity in \(SP\times [0,r_P]\).
To conclude the proof of the lemma, from the uniformly continuity of \({\mathcal {H}}\), given \(\epsilon =\frac{1}{2}\) there is \(\delta \) such that
The last inequality implies that
Consequently, if \(\tilde{x}\in B^P_{\delta }(x)\), then there is \(\tilde{t}\) with \(|\tilde{t}|<\delta \), \({v}\in T^1_{x}P\) such that \(\text {exp}^P_{x}\, \tilde{t}{v}=\tilde{x}\). Thus, (A.1) provides the result of the lemma. \(\square \)
Corollary A.2
In the same condition of the Lemma A.1, there is a constant \(\Gamma >1\) such that
Proof
For each \(x\in P\) consider the function \(\Gamma (x)=\displaystyle \sup \nolimits _{y\ne x}\dfrac{d_{P}(x, y)}{d(x, y)}\). We state that there is \(\Gamma >1\) such that \(\Gamma (x)\le \Gamma \) for all \(x\in P\). In fact, by contradiction assume that for each \(n\in {\mathbb {N}}\) there is \(x_n\) such that \(\Gamma _{n}\ge n \). By definition of \(\Gamma (x_n)\) there is \(y_n\ne x_n\) with \(\dfrac{d_{P}(x_n, y_n)}{d(x_n, y_n)}\ge n-1\). Since P is a compact submanifold, then we can assume that \(x_n\) converges to x and \(y_n\) converges to y. From the last inequality, we have that \(x=y\). Therefore, from Lemma A.1, for n large enough \(y_n\in B_{\delta }(x_n)\) and
which provides a contradiction. \(\square \)
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Melo, Í., Romaña, S. Some Rigidity Theorems for Anosov Geodesic Flows in Manifolds of Finite Volume. Qual. Theory Dyn. Syst. 23, 114 (2024). https://doi.org/10.1007/s12346-024-00972-7
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DOI: https://doi.org/10.1007/s12346-024-00972-7