By Rafael O. Ruggiero
Summary. Manifolds without conjugate issues are typical generalizations
of manifolds with nonpositive sectional curvatures. they've got in
common the truth that geodesics are worldwide minimizers, a variational property
of geodesics that's rather certain. The limit at the signal of the
sectional curvatures of the manifold ends up in a deep wisdom approximately the
topology and the worldwide geometry of the manifold, just like the characterization
of greater rank, nonpositively curved areas as symmetric areas. However,
if we drop the assumptions about the neighborhood geometry of the manifold
the learn of geodesics turns into a lot more durable. the aim of this survey
is to offer an outline of the classical idea of manifolds with out conjugate
points the place no assumptions are made at the signal of the sectional
curvatures, because the well-known paintings of Morse approximately minimizing geodesics
of surfaces and the works of Hopf approximately tori with no conjugate points.
We shall express vital classical and up to date functions of many instruments of
Riemannian geometry, topological dynamics, geometric team conception and
topology to review the geodesic
ow of manifolds with no conjugate points
and its connections with the worldwide geometry of the manifold. Such applications
roughly exhibit that manifolds with out conjugate issues are in many
respects with regards to manifolds with nonpositive curvature from the topological
point of view.
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Additional resources for Dynamics and global geometry of manifolds without conjugate points
The solutions J(t), Y (t) are linearly independent, so their Wronskian is a non-zero constant, W (J(t), Y (t)) = J (0)Y (0) − J(0)Y (0) = J (0)Y (0) = Y (0) > 0. Let us consider the function h(t) = h (t) = J(t) Y (t) . Its derivative is 1 1 (J (t)Y (t) − J(t)Y (t)) = 2 W (J, Y ) > 0, Y 2 (t) Y (t) and thus h(t) strictly increasing. Let us suppose by contradiction that the proposition is false. Then there exists a sequence xn → +∞ such that limn→+∞ J(xn ) = c ∈ R. Let an be Weak stability and hyperbolic geometry 57 the sequence given by Y (xn ) = an J(xn ), which is decreasing since an = 1 h(xn ) and h(t) is increasing.
The system ψt is called quasi-Anosov if for every non-zero vector V ∈ T N that is linearly independent from the vector field that is tangent to the flow we get sup t∈R Dψt (V ) = +∞. Anosov systems are quasi-Anosov, but the converse of this assertion is not true in general. 2. Let (M, g) be a compact manifold without conjugate points. If the Green bundles are linearly independent then the geodesic flow is quasiAnosov. 3. Let (M, g) be a compact manifold without conjugate points. Suppose that there exists a geodesic γθ and perpendicular Jacobi field J defined in γ such that J(t) ≤ C for every t ≥ 0.
Let us define a sequence yn , n ∈ N, by the following recursive formula: y0 y1 = = yn .... = x0 U /4 (h(y0 )) ∩ S U /4 (h(yn−1 )) /4 (x1 ) ∩S /4 (xn ). Let us now show that the orbit of pn = h−n (yn ) 2 -shadows the sequence xi , for 0 ≤ i ≤ n. To see this, we prove first that d(hi (pn ), yi ) = d(hi−n (yn ), yi ) < 2 , n0 and that hi (pn ) ∈ U /2 (yi ), for every 0 ≤ i ≤ n. We make a sort of reverse induction. For i = n we have hi (pn ) = hn (pn ) = yn which satisfies the above two properties.