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        "quote": "设 $G$ 是 $( M , g )$ 的等距变换李群，其李代数为左不变向量场构成的李代数。若 $A \\in { \\mathcal { G } }$ 是左不变向量场，则诱导向量场 $X = { \\widetilde { A } }$ 称为 $M$ 上的 <u id=\"concept-19-killing-vector\" class=\"concept-term\" data-concept-identity=\"urn:uuid:ce4e2dec-50cc-4729-bb8e-6003fcf99e35\" data-concept-route=\"/concept-identities#concept-ce4e2dec-50cc-4729-bb8e-6003fcf99e35\">基灵向量(Killing vector)</u>。若 $\\sigma _ { t }$ 是由 $A$ 生成的单参数等距变换群，则由<a href=\"../chapter-15/05-commutators-flows-and-lie-derivatives#equation-15-33\" class=\"cross-reference\" id=\"xref-05-groups-of-isometries-2\" data-reference=\"equation-15-33\">式 (15.33)</a>，我们有",
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        "quote": "Let $G$ be a Lie group of isometries of $( M , g )$ , and its Lie algebra of left-invariant vector fields. If $A \\in { \\mathcal { G } }$ is a left-invariant vector field then the induced vector field $X = { \\widetilde { A } }$ is called a <u class=\"concept-term\" data-concept-identity=\"urn:uuid:ce4e2dec-50cc-4729-bb8e-6003fcf99e35\" data-concept-route=\"/concept-identities#concept-ce4e2dec-50cc-4729-bb8e-6003fcf99e35\" id=\"first-4a92be17c89ec37308eb-en\">Killing vector</u> on $M$. If $\\sigma _ { t }$ is the one-parameter group of isometries generated by $A$ then, by <a href=\"../chapter-15/05-commutators-flows-and-lie-derivatives#equation-15-33\" class=\"cross-reference\" id=\"xref-05-groups-of-isometries-2\" data-reference=\"equation-15-33\">Eq. (15.33)</a>, we have",
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