geometers think of diff eqs in terms of vector fields. functions of various sorts have derivatives which can be viewed as vector fields, and conversely, given a vector field, one can ask whether they arise from differentiating certain special types of functions.
for instance a smooth function defined on the plane has a gradient field, assigning to each point of the plane the gradient of the function at that point. The inverse problem is to look at a vector field in the plane and ask whether it arises as a gradient field for some smooth function, and if so, to find that function.
The solution is found by considering the path integrals of the vector field taken along curves in the plane. By the Fundamental theorem of calculus, the integral of a gradient field along some curve equals the difference in the values of that function at the ends of the path. In particular the integral in this case depends only on the end points of the path and not on the particular trajectory taken by the path in going from the initial point to the terminal point. Indeed, any vector field which has this property, that path integrals over it depend only on endpoints of the paths, is a gradient field, and the function whose gradient the field is, can be computed, up to a constant, by path integration. I.e. since the integral of df along a path from p to q equals f(q)-f(p), given the integral and fixing an arbitrary choice of the value f(p), we can use this formula to define f(q) for every q. The condition of being a gradient is detected locally by the vanishing of the “curl” of the vector field. I.e. a vector field whose curl is zero, is a gradient at least within any disc contained in its domain, but if we range over the entire domain, the function of which it is a gradient may not be well defined on a loop that encloses a point outside the domain. e.g. the vector field dtheta, is defined and has curl zero everywhere away from the origin, but the angle function theta cannot be continuously defined on any loop containing the origin.
This technique gives a fascinating way to study the topology of smooth surfaces. Visualizing a g-holed doughnut say, standing on end, the height function has one max, one min, and 2g saddle points on the doughnut. Thus the gradient field will have 2g + 2 zeroes, since the gradient at any critical point is zero. But using the matrix of second derivatives we can distinguish extrema from saddle points, and assign a sign to these critical points, +1 to a max or min, and -1 to a saddle point. Then the sum of the signed values assigned to these critical points is 2-2g, exactly the euler characteristic of the surface! In fact any general vector field can be used to calculate the euler characteristic, gradient or not. A converse is also true and enables one to classify compact 2 manifolds. Namely on such a manifold (say oriented) there always exists a smooth function with one max one min and a finite number 2g of saddle points. Using flows arising from solving (as in the next paragraph) the differential equation associated to the gradient field, it follows the manifold is homeomorphic to a doughnut with g handles. Even in higher dimensions, a compact oriented manifold with a smooth function having exactly two critical points, is homeomorphic to a sphere, proved by the same method.
Since a more general vector field in the plane usually will not satisfy the criterion of path independence for integrals, we seek a different way to realize them geometrically. Recall that a smooth path in the plane will have a velocity vector at each point. Given a smooth vector field in the plane and a point, we can ask whether there is a path through that point whose velocity vectors agree with the specified vectors at least at points of that path. Not only is this usually true, but it can be done with a whole family of paths, so that the velocity vectors of all the paths in fact fill up a region in the plane and exhaust all the given vectors there.
Thus given a vector field in a region in R^n, we have ways to find either a family of maps R—>R^n, i.e. paths, or in some cases a single map R^n—>R, i.e. a function, whose derivatives give back the vector field. And as discussed above, looking at the "family of paths" solution to a field originally defined as a gradient, illuminates the topology of the manifold.
i have not studied this but just from trying to explain it here the idea seems to have become clear. Imagine a compact 2 manifold with a smooth function having exactly two critical points, both “non degenerate”. Then look at the gradient field of this function. The critical points must be a max and a min, and since they are non degenerate, the gradient vectors all point in radially at the max and all point out radially at the min. Now draw a little disc around each critical point and remove these 2 discs from the manifold. We are left with a manifold with boundary, whose boundary consists of two circles, and a gradient vector field on it which is everywhere non zero, with vectors all pointing away from the min circle and all pointing in toward the max circle. Now fill in the “flow” or path solutions to this differential equation, hence filling the manifold by paths that extend from the lower circle to the upper circle, all disjoint from each other, i.e. parallel, and filling up the surface. What is this surface? It seems persuasive that it is a cylinder. Then what was the original manifold, obtained by replacing the two discs, one at the top and one at the bottom. Surely this is a sphere.
The same argument applies to classify any oriented surface, but we must remove in addition to the two discs at the max and min, also one saddles at each of the other critical points. It is not as easy to picture, but we again obtain for the complement a disjoint union of cylinders, and when we replace not only the two discs, but also the saddles, one can show we get a doughnut with half as many holes as saddles. E.g. a single holed torus is the union of two discs and two saddles and two cylinders. This called “Morse theory”.
Notice how clever is the method: we start from a differential equation whose solution we know as a gradient. I.e. we start from a function, and look at its gradient field. So we know the solution to the equation asking us to find a function whose gradient gives this field. But then we flip the question around and ask instead for a family of paths solving this same differential equation. That solution also exists and gives us a description of the manifold in terms of paths, which helps us better understand the manifold. I.e. giving a function f:M-->R on our surface M, decomposes M as a union of level curves: for each constant a, we have the curve of {all x in M : f(x) = a}. But then if we look at the gradient field of this f, and fill up this gradient field by paths, by get a description of M in terms of maps R^2-->M, which gives a better description of the manifold by a union of "charts".
There are also criteria for determining whether the family of paths satisfying a vector field defines a coordinate system in the plane, in terms of the vanishing of the “lie bracket” of the vector fields, which basically reflects the fact that mixed partials are equal for a function of several variables. I.e. a solution to the problem of realizing a vector field as the velocity vectors of a family of paths, is essentially a map F:R^2—>R^2, such that through each point p = (F(x0),F(y0)), the solution path is the restriction of F to the line (x,y0) obtained by fixing y = y0, and letting x vary. This means the vector field is given by the family of velocity vectors ∂F/∂x. But this function F also defines another velocity vector field, namely ∂F/∂y. So a more refined problem is to give ourselves two vector fields in the plane and ask when there exists a "change of coordinates" map F such that the first vector field is given by ∂F/∂x and the second is given by ∂F/∂y. The equality of mixed partials of F forces in this case the vanishing of the so called :”bracket product” of the two given vector fields, and conversely that is a sufficient condition for the existence if such a solution map F. i hope i got this more or less right, but i am a novice in diff geom.
People are also interested in other partial diff eq’s, and these too can have a geometric interpretation. Laplace’s equation, for instance comes up famously in complex analysis, as satisfied by the real and imaginary parts of holomorphic functions. It also yields, in complex geometry, a refinement of the famous de Rham theory from differential topology, expressing the cohomology of a smooth manifold by means of smooth differential forms. In this theory, which is analogous to that detecting when a vector field is a gradient, each cohomology class is represented by a whole coset of “closed” differential forms, analogous to those with curl zero, modulo the subspace of “exact” forms, those analogous to gradients. But in the presence of a complex structure on our manifold, we can also define “harmonic” forms, those satisfying the Laplace equation. It turns that then in each cohomology class there is a unique closed harmonic form representing that class, modulo exact forms. It is obviously more convenient to be able to work with a specific form rather than a whole coset of them, but I am not an expert.
My own special favorite is the heat equation, for a function F(t,z) of several complex variables, (tjk, zi), 1≤ i,j,k ≤ n, which equates the first derivative of F wrt tjk, to a constant multiple of the second derivative wrt zj,zk. The fundamental solution function F, the famous “theta function” of Riemann, defines a family of hypersurfaces, defined by fixing t and setting F(z,t) = 0 as a function of z, where these hypersurfaces live in a family of n dimensional complex analytic tori, or abelian varieties. The geometry comes from the heat equation, which links the quadratic tangent cones at double points of these hypersurfaces, to tangent hyperplanes in the family parametrizing these complex tori. The most famous such tori are those defined as jacobian varieties of complex curves, or Riemann surfaces, and using the heat equation, one can show that the locus of such jacobians in the moduli space of tori, agrees locally with the locus where the given hypersurface has a fixed dimensional locus of double points. One can also show that the given Riemann surface can in general be cut out in the projective space defined by lines in the tangent space at the origin of the torus, by those quadric hypersurfaces that occur as tangent cones to double points of the hypersurface defined by the theta function, i.e. the curve is determined by its jacobian variety, Torelli’s problem.
any interested party may consult this link:
http://www.numdam.org/item/CM_1990__76_3_367_0.pdf
for more on the geometry of differential equations i strongly recommend any book by v. arnol'd, such as his intro to ordinary diff eq.
https://www.amazon.com/dp/0262510189/?tag=pfamazon01-20
another undergraduate source for using differential equations to do topology is Wallace:
https://www.amazon.com/dp/0486453170/?tag=pfamazon01-20
