zbMATH — the first resource for mathematics

Variational sequences on finite order jet spaces. (English) Zbl 0813.58014
Janyška, Josef (ed.) et al., Differential geometry and its applications. International conference, Brno, Czechoslovakia, 27 Aug. - 2 Sept. 1989. Singapore: World Scientific. 236-254 (1990).
The variational sequence of order $$r$$ is introduced in the following way. Let $$Y$$ be a fibered manifold with base $$X$$ and $$J^{r}Y$$ the $$r$$-jet prolongation of $$Y$$; further $$\Omega^ k_ r$$ denotes the sheaves of $$k$$-forms over $$J^ rY$$ while $$\Omega^ k_{r(c)}$$ stands for their subsheaves of contact. We set $$\theta^ k_ r = d\Omega^{k - 1}_{r(c)} + \Omega^ k_{r(c)}$$ and consider the quotient sheaves $$\Omega^ k_ r/ \theta^ k_ r$$ jointly with the quotient morphisms $$E_ k : \Omega^ k_ r/\theta^ k_ r \to \Omega^{k + 1}_ r/\theta^{k + 1}_ r$$ induced by the exterior derivatives $$d_ k : \Omega^ k_ r \to \Omega^{k + 1}_ r$$.
The quotient sequence: $0 \to R \to \Omega^ 0_ r \to \Omega^ 1_ r/\theta^ 1_ r \to \Omega^ 2_ r/\theta^ 2_ r \to \cdots$ is called the variational sequence of order $$r$$ over $$Y$$. One of the main theorems of the paper states that such a sequence is an acyclic resolution of the constant sheaf $$R$$ over $$Y$$. Explicit expression for $$E_{n + 1}$$ is provided and it is shown that $$E_ n$$ is the Euler- Lagrange mapping in the sense introduced earlier by the same author.
The paper reads well and constitutes a valuable contribution to the calculus of variations on manifolds.
For the entire collection see [Zbl 0777.00040].

MSC:
 5.8e+31 Variational principles in infinite-dimensional spaces
{"ft_lang_label":"__label__en","ft_lang_prob":0.80270755,"math_prob":0.999902,"size":2038,"snap":"2022-05-2022-21","text_gpt3_token_len":626,"char_repetition_ratio":0.12045231,"word_repetition_ratio":0.051779937,"special_character_ratio":0.3263003,"punctuation_ratio":0.14177215,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.999998,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-20T05:50:29Z\",\"WARC-Record-ID\":\"<urn:uuid:431f8a5b-b0e0-4ff4-ad39-76e4faa036d1>\",\"Content-Length\":\"47088\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:13714aac-6012-4471-a39d-a67547e580b8>\",\"WARC-Concurrent-To\":\"<urn:uuid:71adb563-faa9-477d-bf79-9d10087eacce>\",\"WARC-IP-Address\":\"141.66.194.2\",\"WARC-Target-URI\":\"https://zbmath.org/?q=an:0813.58014\",\"WARC-Payload-Digest\":\"sha1:EOS2A7YI2O5N4BQOQWQ4N4VBFHOOZJQI\",\"WARC-Block-Digest\":\"sha1:4NAFG335JE232YCXA5V4IT6A7VZ7SPKV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320301720.45_warc_CC-MAIN-20220120035934-20220120065934-00455.warc.gz\"}"}