The second closed geodesic on Finsler spheres of dimension 𝑛>2

Abstract

We show the existence of at least two geometrically distinct closed geodesics on an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional sphere with a bumpy and non-reversible Finsler metric for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 2 period"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2.</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n &gt; 2.</mml:annotation> </mml:semantics> </mml:math> </inline-formula>