Given a differentiable scalar field $f$ on a Riemannian manifold $X$ (with properties as required) I would like to formulate a homotopy of maximal integral curves $\gamma, \tilde{\gamma}$ of the gradient field $\nabla f$. Edit: If $X$ has a boundary, i.e. $\partial X \neq \emptyset$ then $f$ shall obey the Neumann boundary condition $g(\mathbf{n}, \nabla f) = 0$ on $ \partial X$ with $\mathbf{n}$ denoting the outward-pointing normal and $g$ the Riemannian metric.
I assume the term "maximal integral curve" as defined in http://math.stanford.edu/~conrad/diffgeomPage/handouts/intcurve.pdf, Theorem 5.1.
Edit: For the constructions below I mainly care about the images of $\gamma, \tilde{\gamma}$ so I allow for an arbitrary parametrization of the curves, even if this parametrization does not strictly coincide with the gradient any more (i.e. not in terms of absolute value, but just in terms of direction).
Then, without loss of generality, let $\gamma, \tilde{\gamma}$ be parameterized over the unit interval $I$. Then I mean in particular a homotopy $H$:
\begin{align} \tag{1} H(\,\cdot\,, 0) \quad &= \quad \gamma \\ H(\,\cdot\,, 1) \quad &= \quad \tilde{\gamma} \tag{2}\\ \forall t \in I \colon \; H(\,\cdot\,, t) \quad & \text{is a maximal integral curve of $\nabla f$} \tag{3} \end{align}
The requirement of the intermediate states being also maximal integral curves is not usual for a homotopy AFAIK. Also it would not necessarily require the end points of each curve to be the same.
First bunch of questions: Is $H$ then actually a homotopy or should I call it rather an isotopy or how else? What would be the proper terminology for this definition?
Now what I need is that all curves that are homotopic to an initial maximal integral curve $\gamma_0$ in this sense cross the same level sets of $f$ in the same order. Formally: For every $\gamma$ homotopic to $\gamma_0$ in sense (1)/(2)/(3) there exists a homeomorphism (a reparametrization) $\psi \colon I \longrightarrow I$ such that \begin{equation} \tag{4} f \circ \gamma = f \circ \gamma_0 \circ \psi \end{equation} This seems like a triviality as a levelset cannot abruptly end unless it contains critical points of $f$. But at critical points the maximal integral curves of $\nabla f$ would end and a homotopy could not be continued in first place. Integral curves of $\nabla f$ must cross a level set orthogonally which would not be defined at the boundary of the level set.
Second bunch of questions: Is there a more formal argument for this or a known theorem yielding this? Or how would I prove it? Or do I even overlook something and there is a counter example?
Given an initial maximal integral curve $\gamma_0$ I suppose there is a well-defined subset $U_0$ of the manifold $X$ consisting of points that lie on any integral curve that is homotopic to $\gamma_0$ in the sense of (1)/(2)/(3). I further suppose that $X$ can be split into disjoint regions $U_j$ in this manner. I.e. such that each region consists of points lying on an integral curve homotopic to an initial integral curve $\gamma_j$ but $\gamma_j$ not homotopic to $\gamma_i$ for $i \neq j$ in sense of (1)/(2)/(3).
Third bunch of questions: Is there a proper terminology for this construction, e.g. "homotopy regions" of $\nabla f$ or something (maybe related to the "domain of flow" from the lecture notes linked above)? Is there a reason to assume that $X$ splits into a finite or at least countable number of these regions $U_j$? I guess $X$ would have to be something like a fractal to yield an infinite number of such regions.
Final bunch of questions: At the top I wrote "with properties as required". What properties are required to make the constructions in this question reasonable? I would prefer to not require $X$ to be compact or bounded, nor to expect that the maximal integral curves would be complete. Further I would like to have this for manifolds with or without boundary. Is that feasible? Does $X$ have to be orientable?
Edit: I realized this may require some more regularity assumptions on $f$. In my use case, $f$ arises as a solution of an elliptic differential equation, thus it should obey a unique continuation property, i.e. $\nabla f$ would not vanish on an open set. $\nabla f$ would vanish almost nowhere on $X$. I am not sure if $f$ can be assumed to be analytic or if $X$ can. However, $f$ should be twice continuously differentiable as the differential equation is second order. Further, I suppose one can only have finitely many $U_j$ if the set of critical points of $f$ consist only of finitely many connectivity components. I am not sure what role it would play whether $X$ is bounded or not.
Thanks in advance!
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