definition of Lie bracket (commutator) of \(C^\infty\) vectors fields on \(C^\infty\) manifold with boundary
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of Lie bracket (commutator) of \(C^\infty\) vectors fields on \(C^\infty\) manifold with boundary.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( M\): \(\in \{\text{ the } C^\infty \text{ manifolds with boundary }\}\)
\( (T M, M, \pi)\): \(= \text{ the tangent vectors bundle over } M\)
\( V_1\): \(: M \to T M\), \(\in \{\text{ the } C^\infty \text{ vectors fields on } M\}\)
\( V_2\): \(: M \to T M\), \(\in \{\text{ the } C^\infty \text{ vectors fields on } M\}\)
\(*[V_1, V_2]\): \(= V_1 V_2 - V_2 V_1\), \(\in \{\text{ the } C^\infty \text{ vectors fields on } M\}\)
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Conditions:
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2: Note
Let us see that \([V_1, V_2]\) is indeed a \(C^\infty\) vectors field on \(M\).
Let us see that at each \(m \in M\), \([V_1, V_2]_m\) is indeed a tangent vector, which is about that it is a derivation of \(C^\infty\) functions.
Let \(f \in C^\infty (M)\) be any.
1st, let us understand the meaning of \(V_1 V_2 - V_2 V_1\). By the proposition that any vectors field is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function is \(C^\infty\), \(V_2 (f), V_1 (f) \in C^\infty (M)\), and so, \(V_1 (V_2 (f)) - V_2 (V_1 (f))\) is possible. For \(V_1 (V_2 (f))\) at \(m\), we need \(V_2\) as a vectors field (not just a tangent vector at \(m\)), because we need \(V_2 (f)\) as a function (not just a number at \(m\)), but we need \(V_1\) only as a tangent vector at \(m\), and likewise for \(V_2 (V_1 (f))\), so, we can write \([V_1, V_2]_m\) as \(V_{1, m} V_2 - V_{2, m} V_1\).
\([V_1, V_2]_m (f_1 f_2) = (V_{1, m} V_2 - V_{2, m} V_1) (f_1 f_2) = V_{1, m} V_2 (f_1 f_2) - V_{2, m} V_1 (f_1 f_2) = V_{1, m} (V_2 (f_1) f_2 + f_1 V_2 (f_2)) - V_{2, m} (V_1 (f_1) f_2 + f_1 V_1 (f_2)) = V_{1, m} (V_2 (f_1)) f_2 + V_2 (f_1) V_{1, m} (f_2) + V_{1, m} (f_1) V_2 (f_2) + f_1 V_{1, m} (V_2 (f_2)) - (V_{2, m} (V_1 (f_1)) f_2 + V_1 (f_1) V_{2, m} (f_2) + V_{2, m} (f_1) V_1 (f_2) + f_1 V_{2, m} (V_1 (f_2))) = V_{1, m} (V_2 (f_1)) f_2 - V_{2, m} (V_1 (f_1)) f_2 - (f_1 V_{2, m} (V_1 (f_2)) - f_1 V_{1, m} (V_2 (f_2))) = (V_{1, m} (V_2 (f_1)) - V_{2, m} (V_1 (f_1))) f_2 - (f_1 (V_{2, m} (V_1 (f_2)) - V_{1, m} (V_2 (f_2)))) = (V_{1, m} V_2 - V_{2, m} V_1) (f_1) f_2 - f_1 (V_{2, m} V_1 - V_{1, m} V_2) (f_2) = (V_{1, m} V_2 - V_{2, m} V_1) (f_1) f_2 + f_1 (V_{1, m} V_2 - V_{2, m} V_1) (f_2) = [V_1, V_2] (f_1) f_2 + f_1 [V_1, V_2] (f_2)\).
\([V_1, V_2]\) is \(C^\infty\), by the proposition that any vectors field is \(C^\infty\) if and only if its operation result on any \(C^\infty\) function is \(C^\infty\).
With the Lie bracket, the set of the \(C^\infty\) vectors fields on \(M\), \(\Gamma (T M)\), becomes a \(\mathbb{R}\) Lie algebra: \(\Gamma (T M)\) is an \(\mathbb{R}\) vectors space; for each \(V_1, V_2, V_3 \in \Gamma (T M)\) and each \(r_1, r_2 \in \mathbb{R}\), 1) \([r_1 V_1 + r_2 V_2, V_3] = (r_1 V_1 + r_2 V_2) V_3 - V_3 (r_1 V_1 + r_2 V_2) = r_1 V_1 V_3 + r_2 V_2 V_3 - r_1 V_3 V_1 - r_2 V_3 V_2 = r_1 (V_1 V_3 - V_3 V_1) + r_2 (V_2 V_3 - V_3 V_2) = r_1 [V_1, V_3] + r_2 [V_2, V_3]\) \(\land\) \([V_3, r_1 V_1 + r_2 V_2] = V_3 (r_1 V_1 + r_2 V_2) - (r_1 V_1 + r_2 V_2) V_3 = r_1 V_3 V_1 - r_1 V_1 V_3 + r_2 V_3 V_2 - r_2 V_2 V_3 = r_1 (V_3 V_1 - V_1 V_3) + r_2 (V_3 V_2 - V_2 V_3) = r_1 [V_3, V_1] + r_2 [V_3, V_2]\); 2) \([V_2, V_1] = V_2 V_1 - V_1 V_2 = - (V_1 V_2 - V_2 V_1) = - [V_1, V_2]\); 3) \(\sum_{cyclic} [V_1, [V_2, V_3]] = [V_1, [V_2, V_3]] + [V_3, [V_1, V_2]] + [V_2, [V_3, V_1]] = [V_1, V_2 V_3 - V_3 V_2] + [V_3, V_1 V_2 - V_2 V_1] + [V_2, V_3 V_1 - V_1 V_3] = V_1 (V_2 V_3 - V_3 V_2) - (V_2 V_3 - V_3 V_2) V_1 + V_3 (V_1 V_2 - V_2 V_1) - (V_1 V_2 - V_2 V_1) V_3 + V_2 (V_3 V_1 - V_1 V_3) - (V_3 V_1 - V_1 V_3) V_2 = 0\).
A major reason for our taking \([V_1, V_2]\) instead of just \(V_1 V_2\) is that \(V_1 V_2\) is not any \(C^\infty\) vectors field: it is not any derivation: just taking a \(C^\infty\) function and producing a \(C^\infty\) function does not make it a \(C^\infty\) vectors field.
