In the first part of the talk I will discuss a "holographic" index theorem for compact manifolds with boundary. It relates the index of a boundary value problem to the index of an operator on the boundary. It will then be applied to scalar curvature geometry. We will show a rigidity theorem for scalar curvature on certain warped product spaces. This implies, in particular, Llarull's theorem for the punctured sphere. A version where a lower bound on scalar curvature is replaced by the dominant energy condition for initial data sets is also discussed. This is based on joint work with Simon Brendle, Aaron Chow, Bernhard Hanke, and Yipeng Wang.