- #1

- 560

- 2

## Main Question or Discussion Point

Left invariant fields on a group G satisfies a lie algebra; say we have an n-dimensional Lie algebra for which the fields ##{X_1, \ldots , X_n}## is a basis. Let these satisfy the algebra ##[X_a, X_b] = c_{ab}^c X_c##. Suppose now that we have a Riemannian manifold with killing vectors ##{\xi_1,\ldots, \xi_n}## and let they satisfy the same algebra ##[\xi_a, \xi_b] = c_{ab}^c \xi_c##. Let ##p \in M## and the action of the group G on M be denoted ##g \cdot p##. Then we have the map ##F: TG \to TM## given by

$$X_a \mapsto X_a^{*} := \left. \frac{d}{dt}\right|_{t = 0} e^{t X_a} \cdot p.$$

Is ##X_a^{*}## identical to the killing field ##\xi_a##? If so, how does one prove it?

$$X_a \mapsto X_a^{*} := \left. \frac{d}{dt}\right|_{t = 0} e^{t X_a} \cdot p.$$

Is ##X_a^{*}## identical to the killing field ##\xi_a##? If so, how does one prove it?