Geodesic flow in certain manifolds without conjugate points

Abstract

A complete simply connected Riemannian manifold <italic>H</italic> without conjugate points satisfies the uniform Visibility axiom if the angle subtended at a point <italic>p</italic> by any geodesic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi> γ </mml:mi> <mml:annotation encoding="application/x-tex">\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <italic>H</italic> tends uniformly to zero as the distance from <italic>p</italic> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi> γ </mml:mi> <mml:annotation encoding="application/x-tex">\gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tends uniformly to infinity. A complete manifold <italic>M</italic> is a uniform Visibility manifold if it has no conjugate points and if the simply connected covering <italic>H</italic> satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let <italic>M</italic> be a uniform Visibility manifold, <italic>SM</italic> the unit tangent bundle of <italic>M</italic> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript t"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the geodesic flow on <italic>SM</italic> . We prove that if every point of <italic>SM</italic> is nonwandering with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript t"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript t"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically transitive on <italic>SM</italic> . We also prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M prime"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">M’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a normal covering of <italic>M</italic> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript t"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically transitive on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper M prime"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">SM’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript t"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_t}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically transitive on <italic>SM</italic> .