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Hey chat, I'm studying homogeneous spaces from the book of Arvaniteyergos, but I have a doubt
Consider $M\cong G/H$ a homogeneous reductive ($\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$) space, and $\pi_{\star,e}:\mathfrak{g}\rightarrow T_o(G/H)$ (where $\mathfrak{m}\cong T_o(G/H)$)
Now, given $X\in \mathfrak{g}$, $\pi_{\star}(X)=X^{\star}_o=\pi_{\star}(X_{\mathfrak{m}})$ (the $\mathfrak{m}$ component), and the author says immediately that $X^{\star}$ is a Killing vector field
Can anyone help me understanding why?

Hi chat, I have some problems with the definition of reductive homogeneous space: we say that a homogeneous space $G/K$ is reductive if there exists a subspace $\mathcal{m}$ of $\mathcal{g}$ such that $\mathcal{g}=\mathcal{m}\oplus \mathcal{k}$, and $Ad(k)\mathcal{m} \subset \mathcal{m}$ $\forall k \in K$ (where $\mathcal{m}$ is the Lie algebra of G, $\mathcal{k}$ of K).
I surely didn′t study enough, but I can′t make" "$Ad(k)\mathcal{m} \subset \mathcal{m}$" make sense
Is it right to read this as "Pick the quotient algebra $\mathcal{g}/\mathcal{m}$ , consider all the images of the adjoint m…

Hi chat, I have some problems with the definition of reductive homogeneous space: we say that a homogeneous space $G/K$ is reductive if there exists a subspace $\mathcal{m}$ of $\mathcal{g}$ such that $\mathcal{g}=\mathcal{m}\oplus \mathcal{k}$, and $Ad(k)\mathcal{m} subset \mathcal{m}$ $\forall k \in \mathcal{m}$ (where $\mathcal{g}$ is the Lie algebra of $G$, \mathcal{k} of K)$:
I surely didn't study enough, but I can't make "$Ad(k)\mathcal{m} \subset \mathcal{m}$" make sense
Is it right to read this as "Pick the quotient algebra $\mathcal{g}/\mathcal{m}$, consider all the images of the a…