Geometrical insights on differential equations Hi: I am researching about relationships between Differential Geometry and Differential Equations. I am looking for examples and references of the use of geometric concepts to solve or analyze differential equations.
For example, in Differential Equations With Applications and Historical Notes the author relates the solution to the Brachistochrone with the Snell law and the behavior of light, providing a very valuable intuition of the form of the solution.
I know a lot of geometric concepts are used in this field: orbits, symmetries... I am looking for especially creative or illuminating examples, especially if they had historical significancy.
For example, in this question a geometric non-obvious intuition for an important theorem in statistics is provided. Another example is the counterexample to Poincaré-Bendixson Theorem counterexample in the torus, where the geometry of the torus is used to construct non-periodic yet recurrent orbits.
Examples and references of relationships in the other direction are also welcome: for example, Picard–Lindelöf theorem can be used to prove every spatial curve is uniquely determined (up to rigid movement) by its curvature and torsion.
Thanks in advance.
 A: As stated in the comment section this is a vast topic. I will just name a few directions that you can explore.
-Degree theory: in finite dimension a map $f:M^n\to N^n$ between closed manifolds of the same dimension has a degree which can be defined as a signed sum of the points in the fiber $f^{-1}(y_0)$ of a regular value $y_0$. This enjoys several properties, for example it does not really depend on $y_0$ and it depends on $f$ only up to homotopy.
This can be generalized (under suitable assumptions)to PDEs once we realize  the solution set of a PDE as the zero locus of a map $F:X\to Y$ where $X, Y$ are Banach spaces/manifolds.
Then if $F$ has positive degree it means that the fiber is non-empty, from this one can conclude the existence of solutions of a perturbation of $F$ or of $F$ itself if $0$ is a regular value.
A book to read about this is Deimling "Nonlinear functional analysis".
-Morse theory: this has been first applied by Marston Morse to prove results about the existence of geodesics on closed manifolds.
We have a functional $E:X\to \mathbb R$ (the energy, e.g. the length of a curve,  $X$ a set of curves) and we are interested critical points of $E$, i.e. solutions to the equation $\nabla E (x) = 0 $. For example the geodesic equation can be formalized in this way (see Critical Curves of the Energy Functional are Geodesics).
Now the idea of Morse is that the flow of the gradient may be used to infer informations about the existence of such critical points. In finite dimension think about a generic  function  $\mathbb S^2\to \mathbb R$, we know that the sum $n_m - n_s + n_M = 2$  where $n_m, n_s, n_M$ are respectively the number of local minimum, saddles, and local maxima.
Consequently if one knows the number of $n_m, n_s$ he can infer the existence of $n_M$ new critical points.
A famous book about this is Milnor's Morse theory.
-h-principle A PDE on a manifold $M$ prescribes an operator $F:J^r(X)\to \mathbb R$ on some jet bundle. Solutions of the PDE must lie in the zero locus of $F$. $J^{r}(X)$ is fibration over $M$, and a solution of the PDE gives a section of $J^{r}(X)$. Unfortunately a section of $J^r(X)$ does not necessarily have to be induced from a function (such sections are called non-holonomic).
The $h$-principle essentially consists in finding sufficient conditions that ensures that you can promote a section of $J^r(X)$ to an holonomic one (hence a real solution).
A book where you can read about this is Eliashberg & Mishachev "introduction to the h-principle". Also see the introduction of https://arxiv.org/pdf/1609.03180.pdf
-Index theory The  Atiyah-Singer index theorem computes the difference between thee dimension of the kernel and the dimension of the cokernel of an elliptic differential operator in purely topological terms.
Studying its proof will give you a lot of tools to understand classic elliptic PDEs and geometry.
A good book about it is Lawson-Michelson's "spin geometry".
