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OK, they say that Cartan didn't have *any* intrinsic definition, but embedded his manifold in a larger cartesian space (which presumably is where the name ‘tangent vector’ comes from, as the vector is literally tangent to the embedded manifold). But they give no credit for the definition of vector as derivation.
And they're not exactly trying to define the tangent space here. Rather, they're starting with a curve through a point and trying to define the derivative. There seems to be no indication that the vector could be defined as an equivalence class of curves itself! In terms of Dan's sibling comment https://mathstodon.xyz/@dpiponi/114337326898112325/, they're identifying B as differential operators but still not figuring out *which* differential operators arise from differentiating curves. (Although they perhaps they do that later on.)
TobyBartels /
npub1kp…gr9jw
2025-04-15 22:17:58
in reply to nevent1q…j5af
Author Public Key
npub1kpf5nh0jvj3ygrlz3zw2s2qvy2dpdp5zdlwnl2nqujugfugumkfqngr9jwPublished at
2025-04-15 22:17:58Event JSON
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