Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

Abstract

Abstract. Let g be a Riemannian metric with ergodic geodesic flow. Then if some metricḡ has the same geodesics (regarded as unparameterized curves) with g, then the metrics are homothetic. If two metrics on a closed surface of genus greater than one have the same geodesics, then they are homothetic.Let g,ḡ be two C 2 -smooth Riemannian metrics on a manifold M n of dimension n ≥ 2. They are projectively equivalent if they have the same geodesics regarded as unparameterized curves. For a given metric g, there always exist trivial examples of projectively equivalent metrics: for any positive constant C, the metric Cg is projectively equivalent to g.Consider the fiberwise-linear mapping G :and the fiberwise-linear mapping A : T M n → T M n given byConsider the characteristic polynomial det(A − λ Id) = c 0 λ n + c 1 λ n−1 + · · · + c n and the mappings S 0 , S 1 , . . . , S n−1 : T M n → T M n given byHere m lies in the set {1, 2, . . . , n} so that k = n − m lies in the set {0, 1, . . . , n − 1}. Consider the functions I 0 , I 1 , . . . , I n−1 : T * M n → R given by the general formulaIn invariant terms, if we identify T M n with T * M n via the metric g, the functions If the geodesic flow of a metric is ergodic, any smooth integral is a function of the Hamiltonian and therefore only trivial examples of projective equivalence are possible. Basically we need not ergodicity but transitivity (= the existence of an

Proof of the projective Lichnerowicz-Obata conjecture

“…The new techniques which allow us to prove the Lichnerowicz-Obata conjecture were introduced in [41,66,67,45,44]: the main observation is that the existence of projective diffeomorphisms allows one to construct commuting integrals for the geodesic flow, see Theorem 5 in Section 2.2. This observation has been used quite successfully in finding topological obstruction that prevent a closed manifold from possessing non-isometric projective diffeomorphisms [43,48,46,49,50].…”

Geodesic rigidity of conformal connections on surfaces

“…It follows that a conformal connection on Σ preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. In particular, as a corollary one obtains that the unparametrised geodesics of a Riemannian metric on Σ determine the metric up to constant rescaling, a result previously proved in [11].…”

Necessary Covariance Conditions for a One-Field Lax Pair

“…The general DT formulas for an operator polynomial in T are given in [11]. We call the operator T a shift operator, but it could be general as defined above.…”

“…There are two types of DT in this case [11], [23], denoted by the superscripts + and −. The DT of the first kind (+) leaves U 0 unchanged.…”

“…Some of them, reviewed in [9], [10], were recently generalized to a wide class of polynomials of automorphisms on a differential ring [11]. Exploring noncommutative coefficients is more complicated but much more rich and promising.…”