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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2024
    • (edited May 15th 2024)

    starting something on isometric immersions

    — mainly I was trying to track down a reference that would clearly state that orthonormal “adapted” or “Darboux” (co)frames (here) always exist locally for an immersion into a Riemannian manifold.

    What I found so far is

    Mastrolia, Rigoli & Setti 2012, p. 33, where this is claimed, but just in passing

    and

    Chen & Giron 2021, Thm. 2.2, where this is stated in the generality of sequences of immersions, which makes it hard to recognize the simple statement behind all the analytic fine-print.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2024

    spelled out of a proof (here) of the existence of local Darboux coframes for an immersion

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2024

    expanded a little more and made adjustments

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2024

    added the observation (here) that the defining property of Darboux co-frames is just what (later) in the super-embedding approach to super pp-branes is called the “embedding condition”.

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2024
    • (edited May 22nd 2024)

    added pointer to (1.12-13) in

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2024

    have added (here) the definition of the “second fundamental form”

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 22nd 2024

    What is the reference Lee 2018?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2024

    Thanks for catching this.

    The link was meant to point to here.

    diff, v17, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTime5 days ago

    added remark (here) on the terminology “second fundamental form” (which leaves room to be further expanded…)

    diff, v20, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTime4 days ago

    added (here) discussion of the second fundamental form in terms of covariant derivatives of and along tangential vectors.

    diff, v23, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTime4 days ago

    also added (here) the coordinate expression and the relation to the Laplace operator characterizing harmonic maps.

    diff, v25, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTime4 days ago

    added (here) a couple of paragraphs on the “tension field” of a Riemannian immersion, and the statement that its vanishing characterizes harmonic maps

    diff, v27, current