Covariant vs absolute derivative

  • Thread starter pmb
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  • #1
pmb
In the online text on differential geometry

http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/pdfs/DiffGeom.pdf [Broken]

The author calls the "derivative along the curve" (aka absolute derivative) the "covariant derivative" which is wrong.

It's on box 8.2 on page 59.

Does anyone else here refer to DP/dtau as the covariant derivative of P?

Pete
 
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Answers and Replies

  • #2
Waner talks about total and partial covariant derivatives on pp. 59-61 and even covariant differentials on p. 62, with no regard to the business of setting new lower indices at all. There is a clue on p. 62, exercise set 8 #10(b), where a contravariant derivative is suggested but not exhibited. This yields some fruit under web search.

There seem to be covariant AND contravariant differential geometries, covariant AND contravariant affine connections, and covariant AND contravariant differentiations afoot. So, I suppose, that means partial and total derivatives of both kinds.

some found links -->

http://emis.bibl.cwi.nl/proceedings/Coimbra99/pdgloja.pdf [Broken]
contravariant connections on poisson manifolds {Fernandes}

http://www.math.toronto.edu/henrique/keio.pdf [Broken]
poisson vector bundles, contravariant connections and deformations {Bursztyn}

The name Izu Vaisman seems to be important.

{SIGH!}, so be the shifting sands of terminology!!

Regards,
 
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