Post A7MnyVSWKIDXKsGbYG by ZevenKorian@mathstodon.xyz
 (DIR) More posts by ZevenKorian@mathstodon.xyz
 (DIR) Post #A7MnUGPrgt07zfbVb6 by ZevenKorian@mathstodon.xyz
       2021-05-18T07:05:32Z
       
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       Let \(\xi\) be a vector field along a smooth mapping \(f\colon N \to P\), and \(f_t\) be an integral curve  of \(\xi\). If \(\omega\) is a differential form on \(P\), then \(f_t^*\omega\) is a differential form on \(N\) (or \(N\times\mathbb{C}\) if you want to keep the parameter as a variable). So far so good.
       
 (DIR) Post #A7MnUGqo4iYHLED1ai by ZevenKorian@mathstodon.xyz
       2021-05-18T07:07:18Z
       
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       Thing is, we now have the Lie derivative of \(\omega\) along \(\xi\), which is defined as\[\left.\frac{d}{dt}\right|_{t=0}f^*_t\omega\] and is again a differential form. I've been thinking how derivating preserves the differential form structure.
       
 (DIR) Post #A7MnUHISPufaiz96gq by ZevenKorian@mathstodon.xyz
       2021-05-18T07:08:37Z
       
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       I imagine it's because \(t\) only affects the coefficients of \(f^*_t\omega\) in a given basis for \(\Omega^k(N)\)?
       
 (DIR) Post #A7MnUHgYyHx5vkQMGO by ZevenKorian@mathstodon.xyz
       2021-05-18T07:23:32Z
       
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       The reason why I'm tinkering about this is because of something called contact geometry. A contact manifold is a smooth manifold \(M^{2n+1}\) where you fix a subbundle \(\Delta\) of \(TM\) of dimension \(2n\). This subbundle is the "kernel" of a 1-form \(\alpha\) such that \(\alpha\wedge d\alpha\neq 0\); that equation is called the non-integrability condition.
       
 (DIR) Post #A7MnUI6nOkw5F6hJ9U by ZevenKorian@mathstodon.xyz
       2021-05-18T07:24:21Z
       
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       Quote-unquote because I'm not sure if there's a well-defined thing such as the kernel of a 1-form, but I think you can easily imagine what I have in mind when I say that.
       
 (DIR) Post #A7MnUIXjmaUEafIp96 by absturztaube@fedi.absturztau.be
       2021-05-18T07:25:13.904261Z
       
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       @ZevenKorian :blobcatgoogly:
       
 (DIR) Post #A7MnmUJujbfQKRViCm by ZevenKorian@mathstodon.xyz
       2021-05-18T07:27:24Z
       
       1 likes, 0 repeats
       
       @absturztaube differential geometry is hard :blobcatlul:
       
 (DIR) Post #A7MnmUoOuG3Nqzm3iy by absturztaube@fedi.absturztau.be
       2021-05-18T07:28:54.101655Z
       
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       @ZevenKorian :blobcatgoogly: obviously, yes. my brain hurts
       
 (DIR) Post #A7MnyVSWKIDXKsGbYG by ZevenKorian@mathstodon.xyz
       2021-05-18T07:29:31Z
       
       1 likes, 0 repeats
       
       @absturztaube :blobcatsnuggle:
       
 (DIR) Post #A7MnyyF3JRfwbVBl5M by absturztaube@fedi.absturztau.be
       2021-05-18T07:31:10.346649Z
       
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       @ZevenKorian :blobsheepsnuggle: