Sharp systolic inequalities for invariant tight contact forms on principal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕊</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -bundles over <mml:math xmlns:mml="http://ww

The systole of a contact form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is defined as the shortest period of closed Reeb orbits of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . Given a non-trivial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕊</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -principal bundle over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕊</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> with total space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> , we prove a sharp systolic inequality for the class of tight contact forms on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> invariant under the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕊</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕊</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.