Vector Fields and Their Flows
A vector field assigns an infinitesimal direction of motion at each point. An integral curve follows those directions, and a flow collects the integral curves for all starting points. The passage from a field to its flow is governed locally by ordinary differential equations; global existence requires additional control.
We first describe vector fields as sections and as derivations on smooth functions. We then examine how smooth maps carry integral curves, and use uniqueness to obtain the composition law for a flow.
1. Vector Fields
Let \(M\) be a smooth Hausdorff, second-countable manifold without boundary; these assumptions remain in force throughout this note. A vector field \(X: M \to TM\) is a continuous assignment \(X(p) = X_p \in T_p M\) such that \(\Pi \circ X = \operatorname{id}_M\), where \(\Pi : TM \to M\) is the projection map. A smooth vector field is a vector field such that the map \(X: M \to TM\) is smooth as a map between the smooth manifolds. In view of Definition 4, a vector field is a section of the tangent bundle.
Given \(M\) is a differentiable manifold, we can write
where \(X^i (p)\) are smooth functions when the field is smooth. As an example, the continuous assignment
determines a vector field on an open neighbourhood \(U\) of \(p \in M\), where \(\left( U, \{x^i\} \right)\) is a smooth chart. This is called the coordinate vector field. Now, we can also speak of vector fields defined “locally” on \(M\). For this, let \(A \subset M\). \(X: A \to TM\) is said to be a vector field along \(A\) if \(\Pi \circ X = \operatorname{id}_A\). If, additionally, for each \(p \in A\), there exists a neighbourhood \(V\) of \(p \in M\) and a smooth vector field \(\widetilde X\) on \(V\) that agrees with \(X\) on \(A \cap V\), then \(X\) is said to be a smooth vector field along \(A\).
The extension lemma lets us replace a field defined near a closed set by a global field with controlled support. This is useful whenever a local construction must be inserted into a global argument.
Lemma 1 (Extension Lemma for Smooth Vector Fields). Let \(M\) be a differentiable manifold, and \(A \subset M\) be a closed subset. Let \(X\) be a smooth vector field along \(A\). Given any open subset \(U \supset A\), there exists a smooth vector field \(\widetilde X\) on \(M\) such that \(\widetilde X|_A = X\) and \(\operatorname{spt} \widetilde X \subset U\), where \(\operatorname{spt} \widetilde X = \overline{\{p \in M: \widetilde X_p \neq 0 \}}\).
Proof of the extension lemma. For each point of \(A\), choose a local extension \(X_i\) on an open set \(V_i\subset U\). Take a locally finite smooth partition of unity subordinate to the cover consisting of these sets and \(M\setminus A\). For a partition function \(\rho_j\) supported in some \(V_{i(j)}\), extend \(\rho_jX_{i(j)}\) by zero outside that set; for functions supported in \(M\setminus A\), use the zero field. Their locally finite sum \(\widetilde X\) is smooth. At a point of \(A\), the latter functions vanish and all local extensions agree with \(X\), so \(\widetilde X=X\). The union of the supports of the former functions is closed, because the family is locally finite, and is contained in \(U\). It contains the support of \(\widetilde X\), giving the required support control. \(\square\)
A particular application of Lemma 1 is to the following result.
Proposition 2. Let \(M\) be a differentiable manifold. For all \(p \in M\) and \(v \in T_p M\), there exists a global, smooth vector field \(X\) on \(M\) such that \(X(p) = X_p = v\).
Proof. Note that the assignment \(p \mapsto v\) is a vector field along the set \(\{p\}\). This assignment is smooth because it can be extended to a constant-coefficient vector field in a coordinate neighbourhood of \(p\). Apply Lemma 1 with \(A = \{p\}\) and \(U = M\) to conclude. ◻
Following convention, if \(M\) is a differentiable manifold, then let \(\mathscr X (M)\) denote the collection of smooth vector fields on \(M\). This is clearly a vector space, with respect to component-wise (pointwise) scalar multiplication and addition. The identity element of this vector space (under addition) is the zero vector field, whose value at each \(p \in M\) is \(0 \in T_p M\). Finally, let \(f \in \mathscr C^\infty (M)\) and \(X \in \mathscr X(M)\). Then, \(f X : M \to TM : p \mapsto (fX)_p = f(p) X_p\) is also a smooth vector field. Algebraically, \(\mathscr X(M)\) is a module over the ring \(\mathscr C^\infty (M)\). With this knowledge, we can express a vector field \(X\) locally in terms of vector fields, as opposed to as vectors at a point, yielding the expression
where \(X^i\) is the \(i\)th component function of \(X\) in the given coordinates.
A frame is a basis of each tangent space chosen as a family of vector fields. It need not arise from coordinate derivatives. To generalize, let \(M\) be a differentiable manifold of dimension \(m\). An ordered \(n\)-tuple of vector fields, \((X_1,\cdots,X_n)\), defined on some subset \(A \subset M\) is said to be linearly independent if \((X_1|_p,\cdots,X_n|_p)\) is a linearly independent \(n\)-tuple in \(T_p M\) at each \(p \in A\). A \(n\)-tuple as above is said to span the tangent bundle if \((X_1|_p,\cdots,X_n|_p)\) spans \(T_p M\) at each \(p \in A\). Accordingly, we say \((X_1|_p,\cdots,X_n|_p)\) defined on \(A \subset M\) is a local frame for \(M\) if it is linearly independent and spans the tangent bundle. Finally, a global frame is a local frame defined on \(A = M\). A frame is said to be smooth if each vector field \(X_i\) is smooth.
A frame on an \(m\)-dimensional manifold has exactly \(m\) members. Given a Riemannian metric \(g\), a frame \((X_1,\ldots,X_m)\) is orthonormal when \(g_p(X_i(p),X_j(p))=\delta_{ij}\). This uses the chosen metric, not an arbitrary identification of \(T_pM\) with Euclidean space. For example, with the Euclidean metric on \(\mathbb R^2\setminus\{0\}\), the following is a smooth global orthonormal frame:
\(X_1\) and \(X_2\) are unit vector fields tangent to radial lines and circles centred at the origin, respectively. For practical purposes, we almost always speak about smooth local frames; smooth global frames are rather rare. We say a differentiable manifold is parallelizable if it admits a smooth global frame.
Example 3. (Parallelizable manifolds.) The coordinate fields frame \(\mathbb R^n\). On a Lie group \(G\), choose a basis \(e_1,\ldots,e_m\) of \(T_eG\) and set \(X_i(g)=d(L_g)_e e_i\). Left translation is a diffeomorphism and depends smoothly on \(g\), so these fields form a smooth global frame. This includes \(\mathbb S^1\) and \(\mathbb T^n\).
Among positive-dimensional spheres, exactly \(\mathbb S^1,\mathbb S^3,\mathbb S^7\) are parallelizable. The exclusion of other dimensions is a deep external input: Adams, Vector fields on spheres, Theorem 1.1 and the preceding construction, p. 603. The positive examples come from multiplication in the complex numbers, quaternions, and octonions: at a unit element \(x\), multiplying an orthonormal imaginary basis by \(x\) gives an orthonormal basis of \(x^\perp=T_x\mathbb S^{d-1}\). The zero-dimensional sphere is also parallelizable, using the empty frame.
Only \(\mathbb S^1\) and \(\mathbb S^3\) among these positive-dimensional spheres are Lie groups. Here is why \(\mathbb S^7\) is excluded. If it were a compact Lie group, averaging an inner product under the adjoint action would give a bi-invariant metric. An abelian connected compact Lie group is covered by its Lie algebra via the exponential map, so cannot be both simply connected and positive-dimensional. In the nonabelian case, the left-invariant three-form \(\omega(X,Y,Z)=\langle X,[Y,Z]\rangle\) is nonzero. Ad-invariance makes it alternating and, since \(\nabla_XY=\frac12[X,Y]\) for a bi-invariant metric, parallel: the three bracket terms in \(\nabla_X\omega\) cancel by Ad-invariance and Jacobi. Thus \(d\omega=\delta\omega=0\). But \(H^3_{\mathrm{dR}}(\mathbb S^7)=0\), so \(\omega=d\alpha\); integration by parts gives \(\|\omega\|_{L^2}^2=\langle\alpha,\delta\omega\rangle_{L^2}=0\), a contradiction. Unit complex numbers and unit quaternions give the remaining Lie groups.
Vector fields are important because they are derivations on \(\mathscr C^\infty (M)\). Let \(f\) be a smooth real-valued function defined on an open subset \(U \subset M\), and \(X \in \mathscr X(M)\). Define a function \(Xf : U \to \mathbb R\) by
In other words, \(Xf\) is obtained from applying the action of \(X_p\) on a real-valued function \(f\). This definition makes sense locally, because the action of a tangent vector is determined by the values of the function in a neighbourhood. In other words, for an open subset \(V \subset U\), we have \((Xf)|_V = X(f|_V)\). Applying this definition multiple times yields the following result.
Proposition 4. Let \(M\) be a differentiable manifold. Then, the following are equivalent:
\(X\) is a smooth vector field.
For every \(f \in \mathscr C^\infty (M)\), \(Xf\) is a smooth function on \(M\).
For every open subset \(U \subset M\) and \(f \in \mathscr C^\infty (U)\), \(Xf\) is a smooth function on \(U\).
Proof of Proposition 4. In local coordinates, \(Xf=\sum_iX^i\partial_if\), so (1) implies (3), which implies (2). Conversely, fix a chart \(U\) and a smaller neighbourhood \(W\) of a point, and choose a smooth cutoff \(\chi\) supported in \(U\) with \(\chi=1\) on \(W\). Extend \(\chi x^i\) by zero to a global smooth function \(f_i\). On \(W\), \(Xf_i=X^i\). Assumption (2) therefore makes every coordinate coefficient smooth near every point, proving (1). \(\square\)
A particular consequence of Proposition 4 is that \(X\) induces an endomorphism on \(\mathscr C^\infty (M)\), defined as \(f \mapsto Xf\). This map is clearly linear over \(\mathbb R\). Furthermore, the Leibniz rule holds: \(X(fg) = f Xg + g Xf\). Any map satisfying these conditions is called a derivation. We can also say that any derivation must be induced by some smooth vector field.
Proposition 5. Let \(M\) be a differentiable manifold. A map \(D : \mathscr C^\infty (M) \to \mathscr C^\infty (M)\) is a derivation if and only if \(Df = Xf\) for some \(X \in \mathscr X(M)\).
Proof of Proposition 5. A smooth vector field is a derivation by the ordinary product rule. Conversely, let \(D\) be an \(\mathbb R\)-linear derivation. The identity \(D1=D(1\cdot1)=2D1\) gives \(D1=0\). If \(h\) vanishes near \(p\), choose a smooth cutoff \(\chi\) equal to one near \(p\) and supported where \(h=0\). Since \(h=(1-\chi)h\), the product rule gives \(Dh(p)=0\). Thus \(Df(p)\) depends only on the germ of \(f\) at \(p\).
In a coordinate ball about \(p\), the fundamental theorem of calculus gives Hadamard’s formula
Locality allows all functions in this identity to be replaced by global cutoff extensions before applying \(D\) at \(p\). The product rule then yields
where the same cutoff is one on a neighbourhood of \(p\). On that neighbourhood each \(a_i\) is smooth. Hence \(X=\sum_i a_i\partial_i\) is a smooth local field, and the identity \(Xf=Df\) makes the fields agree on chart overlaps. They define a global field; testing coordinate functions also proves uniqueness. \(\square\)
We sometimes identify smooth vector fields on \(M\) with derivations of \(\mathscr C^\infty (M)\), using the same letter for both the vector field (thought of as a smooth map \(M \to TM\)), and the derivation (thought of as a linear endomorphism of \(\mathscr C^\infty (M)\)).
We can now discuss more about vector fields in relation to smooth maps. In particular, let \(F : M \to N\) be a smooth map between differentiable manifolds, and \(X \in \mathscr X(M)\). Then, at each \(p \in M\), we can define \((\dif F)_p (X_p) \in T_{F(p)} N\) by applying the differential of \(F\) to \(X_p\). But, this need not be a vector field on \(N\); in fact, in general, it is not. This motivates the following definition.
Definition 6. Let \(F : M \to N\) be a smooth map between differentiable manifolds, \(X \in \mathscr X(M)\), and \(Y \in \mathscr X(N)\). Then, \(X\) and \(Y\) are \(F\)-related if, at each \(p \in M\),
A straightforward calculation shows that, if \(F : \mathbb R \to \mathbb R^2\) defined by \(F(t) = (\cos t, \sin t)\), then \(\od{}{t} \in \mathscr X(\mathbb R)\) is \(F\)-related to \(Y \in \mathscr X(\mathbb R^2)\) defined as \(Y = x \pd{}{y} - y \pd{}{x}\).
Proof. For any \(p \in M\) and any smooth, real-valued function \(f\) defined in a sufficiently small neighbourhood of \(F(p)\), we have
The two expressions are equal for all \(f\) if and only if \(X\) and \(Y\) are \(F\)-related. ◻
We note that \(F\)-related vector fields are unique if \(F\) is a diffeomorphism.
Proof. If \(F\) is a diffeomorphism, we define \(Y\) via
\(Y\) is clearly \(F\)-related to \(X\). Furthermore, \(Y\) is the composition of the maps \(N \xrightarrow{F^{-1}} M \xrightarrow{X} TM \xrightarrow{\dif F} TN\), implying \(Y\) is a smooth map as a composition of smooth maps. The displayed formula is forced at every point because the diffeomorphism is onto, which also proves uniqueness. ◻
Definition 9. The unique vector field obtained in Proposition 8 is called the pushforward of \(X\) by \(F\). This is denoted \(F_* X\). We define this explicitly by \((F_* X)_q = (\dif F)_{F^{-1}(q)} (X_{F^{-1}(q)})\).
Corollary 10. Let \(F : M \to N\) be a diffeomorphism and \(X\in \mathscr X(M)\). For any \(f \in \mathscr C^\infty (N)\),
2. Flows on Vector Fields
Let \(M\) be a differentiable manifold of dimension \(n\), and \(I \subset \mathbb R\) be an interval. Let \(\gamma : I \to M\) be a curve. For all \(t \in I\), it is clear that \(\gamma'(t) \in T_{\gamma(t)} M\). As before, let \(V\) be a vector field along \(M\)—or, equivalently, a section of tangent bundle. This motivates the following definition.
Definition 11. Let \(V\) be a vector field along \(M\). An integral curve of \(V\) is a differentiable curve \(\gamma : I \to M\) such that \(\gamma'(t) = V_{\gamma(t)}\) for all \(t \in I\).
We can write integral curves in local coordinates. Let \(U \subset M\) be an open, smooth domain with coordinates \(\{x^1,\cdots,x^n\}\). Then, the integral curve condition becomes
In other words, locally, integral curves are solutions to a system of ODEs. Standard results in existence and uniqueness of solutions to systems of ODEs yield the following statements.
Proposition 12. For every smooth field \(V\) and \(p\in M\), there is a unique integral curve near time zero with \(\gamma(0)=p\).
Proof. In a chart the equation is \(\dot x^i=V^i(x)\) with \(x(0)=x(p)\). Smoothness makes the right-hand side locally Lipschitz, so the local existence and uniqueness theorem for ODEs applies. Shrink the time interval so the solution stays in the chart, then apply the inverse coordinate map. Chart changes preserve the equation by the chain rule, and uniqueness consequently holds on overlapping coordinate neighbourhoods. \(\square\)
The precise ODE inputs used here and below are local existence/uniqueness and smooth dependence on time and the initial point; see Teschl, Ordinary Differential Equations and Dynamical Systems, Theorems 2.2 and 2.10, pp. 38 and 46–47. Our coordinate right-hand sides are smooth, so they meet both hypotheses.
Proposition 13. If \(\gamma:J\to M\) is an integral curve of \(V\) and \(a\in\mathbb R\), put \(\widetilde J=\{t:at\in J\}\). Whenever this set is nonempty, \(\widetilde\gamma(t)=\gamma(at)\) is an integral curve of \(aV\).
Proof. For \(t\in\widetilde J\) and a smooth local function \(f\),
When \(a=0\) and \(0\in J\), this is the constant curve on all of \(\mathbb R\); if \(0\notin J\), the proposed domain is empty. \(\square\)
Proposition 14. If \(\gamma:J\to M\) is an integral curve of \(V\) and \(b\in\mathbb R\), then \(\widetilde\gamma(t)=\gamma(t+b)\) on \(\widetilde J=J-b\) is an integral curve of \(V\).
Proof. For \(t\in J-b\),
In particular, when \(b\in J\) this curve starts at \(\gamma(b)\). The domain and the argument of \(\gamma\) must use the same sign. \(\square\)
Proposition 15. Let \(M\) and \(N\) be differentiable manifolds. Let \(F: M \to N\) be a smooth map. Then, \(X \in \mathscr X(M)\) and \(Y \in \mathscr X(N)\) are \(F\)-related if and only if \(F\) takes integral curves of \(X\) to integral curves of \(Y\), i.e. if \(\gamma\) is an integral curve of \(X\), then \(F \circ \gamma\) is an integral curve of \(Y\).
Proof. For the “only if” direction, set \(\sigma = F \circ \gamma\). Then,
where the fourth equality follows from \(F\)-relatedness.
For the “if” direction, let \(p \in M\), and \(\gamma : (-\varepsilon,\varepsilon) \to M\) be an integral curve such that \(\gamma(0) = p\). It follows from direct calculations that
i.e. \(X\) and \(Y\) are \(F\)-related. ◻
We now collect the integral curves into maps indexed by time. We do so by first fixing any “time” \(t \in \mathbb R\), and look at the integral curve as it travels across the manifold. To do so, for simplicity, let \(M\) be a differentiable manifold, and \(V\) be a smooth vector field with the property that, for all \(p \in M\), there is a unique integral curve \(\kappa^{(p)} : \mathbb R \to M\) starting at \(p\), i.e. \(\kappa^{(p)}(0)=p\). By fixing a time \(t\), we define the map \(\kappa_t (p) = \kappa^{(p)} (t)\). Doing so for each \(t\), this is a map \(\kappa_t : M \to M\) for all \(t \in \mathbb R\). Note that, by Proposition 14, the map \(t \mapsto \kappa^{(p)}(t+s)\) is an integral curve starting at \(q=\kappa^{(p)}(s)\); since integral curves are assumed to be unique, we must have \(\kappa^{(q)}(t) = \kappa^{(p)}(t+s)\). In other words, \(\kappa_t \circ\kappa_s (p) = \kappa_{t+s} (p)\), and \(\kappa_0 (p) = \kappa^{(p)}(0) = p\). Therefore, the map \(\kappa : \mathbb R \times M \to M\) is an action of the additive group \(\mathbb R\) on \(M\).
Definition 16. A global flow, or one-parameter group action, on differentiable manifold \(M\) is a continuous left-\(\mathbb R\) action on \(M\), i.e. a continuous map \(\kappa : \mathbb R \times M \to M\) such that \(\kappa(t,\kappa(s,p)) = \kappa(s+t,p)\) for all \(s,t \in \mathbb R\), and \(\kappa(0,p) = p\).
Remark 17. Note that the image of the curve \(\kappa^{(p)}\), as defined earlier, is the orbit of \(p\) under the group action.
Definition 18. Let \(\kappa : \mathbb R \times M \to M\) be a smooth global flow on differentiable manifold \(M\). For each \(p \in M\), define \(V_p = \dot \kappa^{(p)} (0)\). The vector field, defined as the assignment \(p \mapsto V_p\), is called the infinitesimal generator of \(\kappa\).
Proposition 19. Let \(\kappa : \mathbb R \times M \to M\) be a smooth global flow on a differentiable manifold \(M\). The infinitesimal generator \(V\) of \(\kappa\) is a smooth vector field on \(M\). Furthermore, each curve \(\kappa^{(p)}\) is an integral curve of \(V\).
Proof. First, we show that \(V\) is smooth. It suffices to show that \(Vf\) is smooth for every smooth real-valued function \(f\), defined on some open subset \(U \subset M\). It follows from direct calculation that for any \(p \in U\),
where the partial derivative with respect to \(t\) is smooth because \(f(\kappa(t,p))\) is a smooth, as a composition of smooth maps. Therefore, \(Vf(p)\) depends smoothly on \(p\), and so is \(V\).
Then, we show \(\kappa^{(p)}\) is an integral curve of \(V\). Fix an arbitrary \(t_0 \in \mathbb R\), and set \(q = \kappa^{(p)} (t_0) = \kappa_{t_0} (p)\). Applying the group law,
Hence, for any smooth real-valued function \(f\) defined in a neighbourhood of \(q\), the action of \(V_q\) on \(f\) is given by
which demonstrates that \(\kappa^{(p)}\) is an integral curve of \(V\), as desired. ◻
Example 20. The vector field \(X = x \pd{}{y} - y \pd{}{x}\) induces the following flow on \(\mathbb R^2\). In coordinates, let \(\gamma(t)= (x(t),y(t))\) denote a curve in \(\mathbb R^2\). For \(\gamma\) to be an integral curve of \(X\), the system of ODEs associated with \(\gamma\) is given by
This system admits the solution
The integral curve is defined for all \(t \in \mathbb R\). Therefore, the global flow generated by \(X\) is given by
For each \(t \in \mathbb R\), note that \(\kappa_t\) rotates the plane through an angle \(t\) about the origin.
While every smooth global flow gives rise to a smooth vector field whose integral curves are precisely the curves defined by the flow, the converse is not true. There are many examples of smooth vector fields whose integral curves are not defined for all \(t \in \mathbb R\). For one example, let \(M=\mathbb R^2\) and \(V = x^2 \pd{}{x}\). Then, the integral curve \(\gamma(t)=(x(t),y(t))\) is given by the pair of equations
With initial value \((a,b)\), the solution is \(x(t)=a/(1-at)\) and \(y(t)=b\), on the connected interval \(1-at>0\) containing zero (all of \(\mathbb R\) if \(a=0\)). In particular, the curve from \((1,0)\) is \((1/(1-t),0)\) on \((-\infty,1)\). It cannot extend through time one: its first coordinate is unbounded there. Thus a smooth field need not generate a global flow, and the time interval of existence can depend on the initial point.
Definition 21. Let \(M\) be a differentiable manifold. Then, a flow domain for \(M\) is an open subset \(\mathscr D \subset \mathbb R \times M\) such that for each \(p\), \(\mathscr D^{(p)} \triangleq \{ t \in \mathbb R : (t,p) \in \mathscr D \}\) is an open interval containing \(0\).
Definition 22. A flow on \(M\) is a continuous map \(\kappa: \mathscr D \to M\) such that \(\kappa(0,p)=p\), and for all \(s \in \mathscr D^{(p)}\) and \(t \in \mathscr D^{\kappa(s,p)}\) such that \(s+t \in \mathscr D^{(p)}\), \(\kappa(t,\kappa(s,p)) = \kappa(t+s,p)\).
Definition 23. A maximal integral curve is an integral curve that cannot be extended to an integral curve on any larger open interval. Similarly, a maximal flow is a flow that admits no extension to a flow on a larger flow domain.
By these definitions, we define the “shifted manifold”
along a flow \(\kappa\). Therefore, \(p \in M_t \iff t \in \mathscr D^{(p)} \iff (t,p) \in \mathscr D\). Furthermore, if \(\kappa\) is smooth, the infinitesimal generator of \(\kappa\) is defined by \(V_p = \dot \kappa^{(p)} (0)\).
Using the differentiation argument from the proof of Proposition 19, we have the following result.
Proposition 24. Let \(\kappa:\mathscr D\to M\) be a smooth flow. Its infinitesimal generator \(V_p=\partial_t\kappa(0,p)\) is smooth, and every \(t\mapsto\kappa(t,p)\) is an integral curve of \(V\).
Proof. Openness of \(\mathscr D\) supplies a product neighbourhood of \((0,p)\), so the smoothness calculation in Proposition 19 applies locally. For \(s\in\mathscr D^{(p)}\), both sides of \(\kappa(t,\kappa(s,p))=\kappa(t+s,p)\) are defined for sufficiently small \(t\). Differentiate at \(t=0\) to obtain \(V_{\kappa(s,p)}=\partial_s\kappa(s,p)\). A merely continuous flow does not justify these derivatives. \(\square\)
Theorem 25. (Fundamental theorem of flows.) Every smooth vector field \(V\) has a unique smooth maximal flow \(\kappa:\mathscr D\to M\) with generator \(V\). Write \(I_p=\mathscr D^{(p)}\) and \(M_t=\{p:(t,p)\in\mathscr D\}\). Then:
- \(\kappa^{(p)}:I_p\to M\) is the unique maximal integral curve from \(p\).
- For \(s\in I_p\) and \(q=\kappa(s,p)\), one has the exact equality \(I_q=I_p-s\) and \(\kappa(t,q)=\kappa(t+s,p)\) on this interval.
- \(M_t\) is open and \(\kappa_t:M_t\to M_{-t}\) is a diffeomorphism with inverse \(\kappa_{-t}\).
Proof. Local uniqueness implies that two integral curves with the same initial value agree wherever both are defined: the equality set in their common interval is closed by continuity and open by local uniqueness. Union all intervals of solutions from \(p\) and glue these compatible curves. This gives a maximal open interval \(I_p\) containing zero and a unique maximal curve \(\gamma_p\). Put \(\mathscr D=\{(t,p):t\in I_p\}\) and \(\kappa(t,p)=\gamma_p(t)\).
To prove that \(\mathscr D\) is open and \(\kappa\) is smooth, fix \((t_0,p_0)\in\mathscr D\). The compact trajectory segment between times zero and \(t_0\) is covered by coordinate neighbourhoods in which the local ODE solution is smooth in time and its starting point. Divide the segment into finitely many sufficiently short time steps, each lying in one such neighbourhood. After shrinking a neighbourhood of \(p_0\), continuity of each local solution ensures that its endpoint remains in the starting neighbourhood for the next step. Compose these finitely many smooth solution maps; the final step can vary in a small interval about its prescribed length. This constructs a smooth solution for all \((t,p)\) in a product neighbourhood of \((t_0,p_0)\). Uniqueness identifies it with \(\kappa\). The same construction with negative steps covers \(t_0<0\).
For \(s\in I_p\), the shifted curve \(t\mapsto\gamma_p(t+s)\) starts at \(q=\gamma_p(s)\). Uniqueness makes it the restriction of \(\gamma_q\), giving \(I_p-s\subset I_q\). The shifted curve is itself maximal: an extension would, after translating time back by \(s\), extend \(\gamma_p\). Thus the inclusion is equality and the flow law follows. Its derivative at zero is \(V\).
Openness of \(\mathscr D\) gives openness of its slices \(M_t\). The interval identity shows that \(q=\kappa_t(p)\) lies in \(M_{-t}\), and the flow law gives \(\kappa_{-t}(q)=p\); the converse follows with \(-t\). Both maps are smooth restrictions of \(\kappa\), proving the diffeomorphism assertion. Finally, any other smooth flow with generator \(V\) consists of integral curves by Proposition 24, hence is a restriction of \(\kappa\). Maximality forces equality. \(\square\)
A useful completeness criterion. A maximal integral curve cannot remain in a fixed compact set as it approaches a finite endpoint of its time interval. Indeed, finitely many local ODE neighbourhoods of that compact set give a uniform positive time of existence from every point in it; start sufficiently close to the endpoint and use uniqueness to extend the curve past it. Thus every smooth field on a compact manifold is complete. More generally, a compactly supported field is complete: outside its support the constant curve is the unique solution, so a nonconstant curve cannot pass through such a point and remains in the compact support.
The flow whose existence and uniqueness are asserted in the fundamental theorem is called the flow generated by \(V\), or simply by the flow of \(V\). The term “infinitesimal generator” comes from the following heuristic: in a smooth chart, a good approximation to an integral curve can be obtained by composing many small straight-line motions, with the direction and length of each motion determined by the value of the vector field at the point arrived at in the previous step. This interpretation explains the name: the vector field specifies the velocity of the flow at time zero, and the group law transports that information to other times.
Reading: Carmo (1992), Riemannian geometry.