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We prove solenoidal injectivity for the geodesic X-ray transform of tensorfields on simple Riemannian manifolds with $C^{1,1}$ metrics and non-positivesectional curvature. The proof of the result rests on Pestov energy estimatesfor a transport equation on the non-smooth unit sphere bundle of the manifold.  Our low regularity setting requires keeping track of regularity and makinguse of many functions on the sphere bundle having more vertical than horizontalregularity. Some of the methods, such as boundary determination up to gauge andregularity estimates for the integral function, have to be changedsubstantially from the smooth proof. The natural differential operators such ascovariant derivatives are not smooth.

Tensor tomography on negatively curved manifolds of low regularity