On unions of geodesics and projections of invariant sets

Let $M$ be a $d$-dimensional complete Riemannian manifold and let $\pi :SM\to M$ denote the canonical projection from the unit tangent bundle. We prove that if $E\subset SM$ is a set that invariant under the geodesic flow with Hausdorff dimension ${\dim }_{\mathcal{H}}E\ge 2(k-1)+1+\beta$ for some integer $1\le k\le d-1$ and some $\beta \in [0,1]$, then the projection $\pi (E)$ satisfies ${\dim }_{\mathcal{H}}\pi (E)\ge k+\beta$. In other words, this yields a lower bound on the Hausdorff dimension of unions of geodesics in $M$. Our theorem extends a result of J. Zahl concerning unions of lines in ${\mathbb{R}}^d$. The proof relies on the transversal property of geodesics, an appropriate $(k+1)$-linear curved Kakeya estimate, and the Bourgain-Guth argument.