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An element of the tangent bundle $T M$ of a manifold is called a "(tangent) vector". An element of its dual $T^* M$ is called a "covector" or a "1-form". An element of the exterior square $\Lambda^2(T M)$ is called a "bivector", and an element of $\Lambda^2(T^*M)$ is called a "2-form". More generally, elements of the various tensor powers of $T M$ and $T^*M$ are called "tensors".

Is there a name for an element of an iterated tangent bundle $T^k M = T(T(\cdots (T M)))$?

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  • $\begingroup$ I'd call them something like "n-fold variations". $\endgroup$
    – Ryan Budney
    May 21, 2014 at 5:55
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    $\begingroup$ Shouldn't we reserve $1$-form to mean a global section of $T^{\ast}$ rather than a point in its total space? $\endgroup$
    – Qiaochu Yuan
    May 21, 2014 at 6:01
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    $\begingroup$ @Qiaochu: people already use the term 1-form for elements of a dual vector space, so global section or not, the terminology is in use. And Paul: what's the distinction between higher-order and iterated tangent bundles? $\endgroup$
    – Ryan Budney
    May 21, 2014 at 6:41
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    $\begingroup$ @PaulReynolds is the second order tangent bundle really a vector bundle? I thought that it wasn't (even though each fiber is a vector space) since jets don't transform like vectors. $\endgroup$
    – Mike Shulman
    May 21, 2014 at 13:04
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    $\begingroup$ FWIW, if we have to invent a name, the best I've thought of so far is "flare" -- like a jet, but less unidirectional. $\endgroup$
    – Mike Shulman
    May 21, 2014 at 13:58

1 Answer 1

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I called them "expansion" because they act on functions on $M$ by (Taylor) expanding them. This is a generalization of derivation. See 37.6 of

  • Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993) pdf.

Perhaps you need to skim the whole chapter VIII.

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