Two real-analytic Finsler metrics on a closed surface of negative Euler characteristic are projectively equivalent if and only if they differ by a positive scaling plus a closed 1-form.
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Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic
Two real-analytic Finsler metrics on a closed surface of negative Euler characteristic are projectively equivalent if and only if they differ by a positive scaling plus a closed 1-form.