It's essential in computer graphics or in computational geometry and in simulation science, where I can speak from experience - these are more intersectional topics than pure CS I think but I'd say it's applied CS.
E.g. in finding faster ways to do Ray/surface intersections (if we're talking about actually industrially useful geometry like all kinds of splines and not just triangle meshes), differential geometry is essential - even with triangle meshes you can apply it in normal and curvature estimation. Differential equations and integrating them enter the picture if you want to find the shortest way from one surface point to another along the arbitrarily shaped surface.
With simulations differential equations are everywhere because any physical system as a function of space (and time) is a collection of differential equations that you need to solve.
Earth-movers distance and the Wasserstein metric have recently got attention again, its original relevance was in the Monge-Ampere problem, how to distribute a continuous distribution of 'heaps' of some kind into a distribution of 'sinks' with the least amount of total distance moved. Which is a nonlinear partial differential equation to solve in two dimensions.
We need to apply numerical methods, nonlinear optimization to solve such problems and CS is a part of doing that quickly. Because there are no general closed form solutions for most of the systems of differential equations or there we need algorithms to solve them approximately.
E.g. in finding faster ways to do Ray/surface intersections (if we're talking about actually industrially useful geometry like all kinds of splines and not just triangle meshes), differential geometry is essential - even with triangle meshes you can apply it in normal and curvature estimation. Differential equations and integrating them enter the picture if you want to find the shortest way from one surface point to another along the arbitrarily shaped surface.
With simulations differential equations are everywhere because any physical system as a function of space (and time) is a collection of differential equations that you need to solve.
Earth-movers distance and the Wasserstein metric have recently got attention again, its original relevance was in the Monge-Ampere problem, how to distribute a continuous distribution of 'heaps' of some kind into a distribution of 'sinks' with the least amount of total distance moved. Which is a nonlinear partial differential equation to solve in two dimensions.
We need to apply numerical methods, nonlinear optimization to solve such problems and CS is a part of doing that quickly. Because there are no general closed form solutions for most of the systems of differential equations or there we need algorithms to solve them approximately.