Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A translation surface is a Riemann surface equipped with a holomorphic 1-form $\omega$ and a Riemannian metric $g=\omega \bar \omega$ with conical singularities. It is well-known that there exists closed regular geodesics, i.e., those not going through singular points, but usually shortest curves in a given homotopy class pass through the singular points.

Question 1: Is the first homology group generated by regular closed geodesics (at least over $\mathbb Q$)? I am mainly interested in the case of genus $5$ surfaces with $\omega$ having four distinct double zeros.

Question 2: If question 1 has a negative answer, I would like to know if there is another way to determine the periods of $\omega$ in terms of the lengths of the regular closed geodesics $\gamma$ and their angles (defined by the (constant) modulus of $\omega(\gamma')$).

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.