Isometry of a semi-Riemannian manifold

Let $(𝑀, 𝑔)$ be a semi-Riemannian $𝐶 𝛼$ -manifold. An isometry $𝜑 \in I s o (𝑀, 𝑔)$ is a $𝐶 𝛼$ -diffeomorphism $𝜑 : 𝑀 \to 𝑀$ which preserves the metric, #m/def/geo/diff i.e. the pullback $𝜑 * 𝑔 𝑎 𝑏$ satisfyies $𝜑 * 𝑔 𝑎 𝑏 = 𝑔 𝑎 𝑏$ . The isometries form a Lie group $I s o (𝑀, 𝑔)$ , whose corresponding Lie algebra $𝔦 𝔰 𝔬 (𝑀, 𝑔)$ consists of Killing fields.

Proof

#missing/proof

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