  Given a metric space $X$ and a subspace $A\subset X$, we prove $A$ can
generate various algebraic elements in persistent homology of $X$. We call such
elements (algebraic) footprints of $A$. Our results imply that footprints
typically appear in dimensions above the dimension of $A$. Higher-dimensional
persistent homology thus encodes lower-dimensional geometric features of $X$.
  We pay special attention to a specific type of geodesics in a geodesic
surface $X$ called geodesic circles. We explain how they may generate
non-trivial odd-dimensional and two-dimensional footprints. In particular, we
can detect even some contractible geodesics using two- and three-dimensional
persistent homology. This provides a link between persistent homology and
length spectrum in Riemannian geometry.
