Math
Higher order terms of $\operatorname{grad}$ on a perturbed Morse flowline
Step-by-step mathematics solution: Higher order terms of $\operatorname{grad}$ on a perturbed Morse flowline
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1. What the question is asking (in plain language)
We have a Morse gradient flow on a Riemannian manifold ((M,g))
[ \frac{d u}{ds}+ \operatorname{grad} f(u)=0 , \qquad L:=\frac{d}{ds}+ \operatorname{grad} f . ]
A pre‑glued trajectory is written as
[ u(s)=\beta_{-}(s)\bigl(u_{-}(s)+\psi_{-}(s)\bigr) +\beta_{0}(s)\bigl(u_{0}(s)+\psi_{0}(s)\bigr) +\beta_{+}(s)\bigl(u_{+}(s)+\psi_{+}(s)\bigr), ]
where
- the three “building blocks’’ (u_{\star}) are exact gradient flow lines,
- (\psi_{\star}) are small perturbations (sections of (u_{\star}^{*}TM)) that lie in the (L^{2}{1})–orthogonal complement of (\ker D{u_{\star}}),
- (\beta_{\star}) are smooth cut–off functions that add up to (1).
When we apply the operator (L) to (u) we obtain
[ L(u)=\beta_{+}\Theta_{+}(\psi_{+},\psi_{0}) +\beta_{0}\Theta_{0}(\psi_{+},\psi_{0},\psi_{-}) +\beta_{-}\Theta_{-}(\psi_{0},\psi_{-}), ]
and each (\Theta_{\star}) contains a linear part (the linearisation (D_{u_{\star}}\psi_{\star}) and the derivative of the cut‑offs) plus a remainder (H(\psi_{\star})).
The question is:
What exactly is the “higher–order term’’ (H(\psi_{\star})) that comes from expanding (\operatorname{grad} f(u_{\star}+\psi_{\star})) beyond the linear term?
We have to write down the Taylor expansion of the vector field (\operatorname{grad} f) in a covariant way, keep all quadratic (and higher) contributions, and identify them with the symbol (H(\psi_{\star})).
2. Full derivation (no steps omitted)
2.1 Linearisation of the gradient flow equation
Fix a smooth flow line (u\colon\mathbb R\to M) satisfying
[ \partial_s u + \operatorname{grad} f(u)=0 . \tag{2.1} ]
Let (\psi\in\Gamma(u^{}TM)) be a section (the perturbation).
Write a *perturbed curve using the exponential map:
[ \widetilde u(s)=\exp_{u(s)}\bigl(\psi(s)\bigr). \tag{2.2} ]
The gradient flow equation for (\widetilde u) reads
[ \partial_s\widetilde u + \operatorname{grad}f(\widetilde u)=0 . \tag{2.3} ]
We expand the left‑hand side in powers of (\psi).
Recall two standard formulas for the exponential map (see e.g. Klingenberg, Riemannian Geometry):
- For a vector field (X) along (u),
[ \frac{D}{ds}\exp_{u}\bigl(X\bigr) = \bigl( D_{u}X \bigr) + O(|X|^{2}), \tag{2.4} ]
where (D_{u}X:=\nabla_{\partial_s}X+\nabla_{X}\operatorname{grad}f(u)) is the linearisation of the gradient flow operator.
- For any smooth vector field (Y) on (M),
[ Y\bigl(\exp_{u}(X)\bigr) = Y(u) + \nabla_{X}Y(u) + \frac12 \nabla^{2}_{X,X}Y(u) + O(|X|^{3}). \tag{2.5} ]
In (2.5) we use the covariant Taylor expansion: (\nabla_{X}Y) is the directional derivative of the vector field (Y) along the vector (X); (\nabla^{2}_{X,X}Y) is the second covariant derivative, i.e.
[ \nabla^{2}{X,X}Y := \nabla{X}\bigl(\nabla_{X}Y\bigr)-\nabla_{\nabla_{X}X}Y . ]
Apply (2.5) with (Y=\operatorname{grad}f) and (X=\psi(s)). Because (\operatorname{grad}f) is a vector field, we obtain
[ \operatorname{grad}f\bigl(\exp_{u}\psi\bigr) = \operatorname{grad}f(u) + \nabla_{\psi}\operatorname{grad}f(u) + \frac12 \nabla^{2}_{\psi,\psi}\operatorname{grad}f(u) + O(|\psi|^{3}). \tag{2.6} ]
Now substitute (2.4) and (2.6) into (2.3):
[ \begin{aligned} 0 &= \partial_s\bigl(\exp_{u}\psi\bigr)
- \operatorname{grad}f\bigl(\exp_{u}\psi\bigr) \[4pt] &= \underbrace{\bigl(\partial_s\psi + \nabla_{\psi}\operatorname{grad}f(u)\bigr)}{\displaystyle D{u}\psi} \;+\; \underbrace{\frac{d\beta}{ds}\,u}{\text{cut‑off term (if any)}} \;+\; \underbrace{\frac12 \nabla^{2}{\psi,\psi}\operatorname{grad}f(u)}_{\displaystyle H(\psi)} \;+\; O(|\psi|^{3}) . \end{aligned} \tag{2.7} ]
The first bracket is exactly the linearised operator
[ D_{u}\psi := \partial_s\psi + \nabla_{\psi}\operatorname{grad}f(u) . \tag{2.8} ]
All remaining terms are quadratic or higher in (\psi).
Hence we define the higher‑order remainder by
[ \boxed{ H(\psi)\;:=\; \frac12 \,\nabla^{2}_{\psi,\psi}\operatorname{grad}f\bigl(u\bigr) \;+\; O\bigl(|\psi|^{3}\bigr) . } \tag{2.9} ]
Because we work in a Euclidean chart around each critical point (the paper assumes a flat metric in a (2\epsilon)-ball), the curvature terms that would appear in the full covariant expression disappear, and the above formula reduces to the ordinary Euclidean Taylor expansion.
2.2 Explicit coordinate form (for the reader’s intuition)
Pick normal coordinates ((x^{1},\dots,x^{n})) centred at a critical point (p).
In these coordinates the Levi‑Civita connection vanishes at the centre, and the metric is the identity matrix up to (O(|x|^{2})).
Write (\psi = (\psi^{1},\dots,\psi^{n})) and (\operatorname{grad}f = (\partial_{1}f,\dots,\partial_{n}f)).
Then
[ \begin{aligned} \bigl(\operatorname{grad}f\bigr)^{i}\bigl(u+\psi\bigr) &= \partial_{i}f(u)
- \sum_{j}\partial_{j}\partial_{i}f(u)\,\psi^{j}
- \frac12\sum_{j,k}\partial_{j}\partial_{k}\partial_{i}f(u)\, \psi^{j}\psi^{k}
- O(|\psi|^{3}) . \end{aligned} ]
The first two terms give (\operatorname{grad}f(u)+\nabla_{\psi}\operatorname{grad}f(u)).
The quadratic part is exactly
[ \bigl(H(\psi)\bigr)^{i} = \frac12\sum_{j,k} \partial_{j}\partial_{k}\partial_{i}f(u)\, \psi^{j}\psi^{k} \;+\; O(|\psi|^{3}) . \tag{2.10} ]
Thus, in Euclidean coordinates the abstract covariant expression (2.9) becomes the familiar third‑order partial‑derivative term of the Taylor series.
2.3 Plugging the expansion into the three pieces (\Theta_{\pm},\Theta_{0})
Recall the definitions from the question:
[ \begin{aligned} \Theta_{+}(\psi_{+},\psi_{0}) &= D_{u_{+}}\psi_{+} + \frac{d\beta_{0}}{ds}\,(u_{0}+\psi_{0}) + H(\psi_{+}),\[4pt] \Theta_{0}(\psi_{+},\psi_{0},\psi_{-}) &= D_{u_{0}}\psi_{0} + \frac{d\beta_{+}}{ds}\,\exp_{u_{+}}(\psi_{+}) + \frac{d\beta_{-}}{ds}\,\exp_{u_{-}}(\psi_{-}) + H(\psi_{0}),\[4pt] \Theta_{-}(\psi_{0},\psi_{-}) &= D_{u_{-}}\psi_{-} + \frac{d\beta_{0}}{ds}\,(u_{0}+\psi_{0}) + H(\psi_{-}) . \end{aligned} ]
The only place where the higher–order term appears is the \(H(\psi_{\star})\) written above.
Because each (u_{\star}) already satisfies the unperturbed flow equation, the linear part (\partial_{s}u_{\star}+ \operatorname{grad}f(u_{\star})) vanishes, leaving the three displayed terms as the complete expansion.
Thus the expanded version of (H) is precisely
[ \boxed{ H(\psi_{\star})(s) \;=\; \frac12\,\nabla^{2}{\psi{\star}(s),\,\psi_{\star}(s)}!\bigl(\operatorname{grad}f\bigr) \bigl(u_{\star}(s)\bigr) \;+\; O!\bigl(|\psi_{\star}(s)|^{3}\bigr) . } \tag{2.11} ]
If one wishes to keep the cubic and higher terms explicitly, write
[ H(\psi_{\star}) =\sum_{k\ge 2} \frac{1}{k!} \nabla^{k}{\underbrace{\psi{\star},\dots,\psi_{\star}}{k}} \bigl(\operatorname{grad}f\bigr)(u{\star}) . \tag{2.12} ]
In the flat chart the covariant derivatives become ordinary partial derivatives, so (2.12) is exactly the usual multivariate Taylor series of the
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