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Suppose a nonempty compact $(n+1)$-manifold with a volume form has $n$ pointwise independent volume-preserving vector fields that commute and are tangent to its boundary. If restriction $H_{\mathrm{dR}}^1(M)\to H_{\mathrm{dR}}^1(\partial M)$ is injective, the boundary must have at least two connected components.

Indeed $\eta ={\iota }_{X_1}\cdots {\iota }_{X_n}\omega$ is nowhere zero. Cartan's magic formula and commutation show $d\eta =0$, while boundary tangency gives $j^{\ast }\eta =0$. Injectivity in de Rham cohomology makes $\eta =df$. On a connected boundary $f$ is constant; a maximum or minimum differing from that boundary value would be an interior critical point, contradicting $df\ne 0$. Empty boundary is also impossible.

A smooth gradient flow confined to a compact invariant sublevel set has finite total integral of $\Vert \nabla V{\Vert }^2$ when $V$ is bounded below. Compactness supplies uniform continuity of that squared norm along the trajectory, so it tends to zero. Every accumulation point is a critical point. If the trapped component contains exactly one critical point, the trajectory converges to it. This supplies a direct convergence proof in place of an inference solely from an energy-contour picture.

On a positive-dimensional Riemannian manifold, equality means equality of the two covariant derivatives at $p$ for every pair of vector fields. A critical point of $\sigma$ annihilates the conformal correction. Conversely, writing that correction as $A$, the identity $g(A(v,v),v)=g(v,v)d\sigma (v)$ for all tangent vectors forces $d\sigma =0$ if $A=0$. A smooth function on a nonempty compact boundaryless manifold has a critical point, so conformally related connections coincide somewhere. Compact manifolds with boundary need not have one.

For $E(\gamma )=\frac{1}{2}{\int }_0^1|\dot{\gamma }{|}^{2}dt$ and variation vector field $V$, the first variation is $[⟨V,\dot{\gamma }⟩{]}_0^1-{\int }_0^1⟨V,D_t\dot{\gamma }⟩dt$. For fixed endpoints the boundary term vanishes, so the critical points of the energy are exactly affinely parametrized geodesics.

A level curve is the planar level set of a scalar function. Where the gradient does not vanish, the implicit function theorem makes it locally a smooth curve. At a critical point, branches may cross, narrow into a cusp or shrink to a point, so a plot should be interpreted with the local critical-point analysis.

The endpoint of a gas-liquid phase coexistence curve, where the distinct liquid and gas states merge. For a smooth fluid equation of state, the critical isotherm has $p_v=p_{vv}=0$ at this point; this thermodynamic meaning differs from a stationary critical point of an arbitrary function.

A Morse-Bott critical manifold is a smooth submanifold $C$ of critical points such that the kernel of the Hessian matrix at every $x\in C$ is exactly $T_{x}C$. The Hessian is therefore nondegenerate in normal directions. Repeated based great-circle geodesics on a round sphere form such critical manifolds, parametrized by their unit initial directions.

A Morse function is a smooth real-valued function whose critical points are all nondegenerate. Near a critical point of index $k$, the Morse lemma gives coordinates in which the function is a constant minus $k$ squares plus the remaining positive squares.

Morse theory relates the critical points of a Morse function to changes in the homotopy type of its sublevel sets. Crossing a nondegenerate critical value attaches a cell or handle of dimension equal to the Morse index. The theory also applies to geodesic energy through broken-geodesic approximations.