By Rafael O. Ruggiero
Summary. Manifolds with out conjugate issues are usual generalizations
of manifolds with nonpositive sectional curvatures. they've got in
common the truth that geodesics are international minimizers, a variational property
of geodesics that's relatively specified. The restrict 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 tougher. the aim of this survey
is to provide an outline of the classical thought of manifolds with no 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 out conjugate points.
We shall exhibit vital classical and up to date functions of many instruments of
Riemannian geometry, topological dynamics, geometric workforce conception and
topology to check the geodesic
ow of manifolds with no conjugate points
and its connections with the worldwide geometry of the manifold. Such applications
roughly express that manifolds with out conjugate issues are in many
respects as regards to manifolds with nonpositive curvature from the topological
point of view.
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Additional info for Dynamics and global geometry of manifolds without conjugate points
3). So the semi-conjugacy between fδ and f is in fact a conjugacy as we wished to show. 4. An Anosov diffeomorphism f : M −→ M defined on a compact manifold M is C 1 structurally stable. Proof. We just outline the proof of this famous result. We can show that if a diffeomorphism h : M −→ M is C 1 sufficiently close to f , then h is Anosov (see for instance ). 2 to h to conclude that h is conjugate to f . Since this happens in an open C 1 neighborhood of f , then f is C 1 structurally stable as we claimed.
The well known Anosov linear automorphisms of the torus illustrate very well the invariant manifold theorem. The linear map f˜ : R2 −→ R2 , f˜(x, y) = (2x + y, x + y) is a symmetric, measure preserving map which preserves as well the lattice of points with integer coordinates. So by means of the covering map from R2 onto T 2 we can project f˜ into a diffeomorphism f : T 2 −→ T 2 . Its differential at every point has two eigenvalues λ < 1 < µ, which yields that f is Anosov. The eigenvectors of Dp f determine the directions of the stable and unstable subspaces at p, and the eigenvalues give the contraction of stable vectors and the expansion of unstable vectors under the action of Dp f .
For every V ∈ Tγ(0) M that is perpendicular to γ (0), the limit lim JT (t)(V ) = JVs (t) T →+∞ exists for every t ∈ R, and it is a perpendicular Jacobi field with JVs (0) = V , 3. Analogously, the limit limT →−∞ JT (t)(V ) = JVu (t) exists, and it is a perpendicular Jacobi field with JVu (0) = V , s JVu (t) never vanish 4. The Jacobi fields √ JV (t), √ if V = 0, and we have s s JV (t) ≤ K0 JV (t) , JVu (t) ≤ K0 JVu (t) , for every t ∈ R. 52 Rafael O. Ruggiero The main idea of the proof is a comparison argument between the solutions of the Riccati equation of (M, g) along γ and the Riccati equation U (t) + U 2 (t) − K0 = 0, where K0 = K0 I, I is the identity matrix.
Dynamics and global geometry of manifolds without conjugate points by Rafael O. Ruggiero