The Lie bracket of smooth vector fields

Depends on

• The action of a vector field on smooth functions

Used by

• Two pairs of vector fields can agree at a point and still have different bracket values there Counterexample
• Curvature of a vector-bundle connection Definition
• Curvature of an affine connection Definition
• Involutive distributions Definition
• Lie bracket on the tangent space of a Lie group Definition
• Principal connections and vertical curvature Definition
• The exterior derivative by the invariant vector-field formula Definition
• A coordinate computation of a nonzero Lie bracket Example
• Gauge action on a trivial real principal bundle and associated line Example
• Great circles as round-sphere geodesics Example
• Parallel transport on the round sphere along the equator Example
• The affine group of the line Example
• The flat connection on a trivial vector bundle Example
• The Heisenberg Lie group and algebra Example
• FALSE: the point values Xₚ and Yₚ determine the bracket value [X,Y]ₚ False statement
• Torsion free means curvature free False statement
• The commutator of vector-field derivations is again a derivation Lemma
• Coordinate formula for the Lie bracket Proposition
• Coordinate vector fields commute Proposition
• Leibniz rules for the Lie bracket with function multiples Proposition
• Level-set distributions are involutive Proposition
• Right-invariant fields carry the opposite Lie bracket Proposition
• The riemannian hessian is symmetric Proposition
• Compatible almost-complex structures and Kähler geometry Remark
• First Bianchi identity Theorem
• Smooth vector fields form a Lie algebra under the Lie bracket Theorem
• The Lie derivative of a vector field equals the Lie bracket Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
• Will J. Merry, Differential Geometry (standard reference, not scraped)