There is other definitions of regular point of a curve, such as in "Elementary Geometry" by John Roe: (7.1.8) Let γ:[ ...
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DefinitionofRegularPoint
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In"AnIntroductiontoManifolds(secondedition)"byLouringW.Tu,regularpointisdefinedas:
Definition8.22.ApointpinNisacriticalpointofFifthedifferential
$F_{∗,p}:T_pN→T_{F(p)}M$
failstobesurjective.ItisaregularpointofFifthedifferential$F_{∗,p}$issurjective.
Underthisdefinition,if$dimN=n,dimM=m$and$nR^2:x=x(t)andy=y(t)doesnothaveanyregularpointsaccordingthethefirstdefinition,butaccordingtotheseconddefinition,itmayhaveregularpointt0ifx'(t0)ory'(t0)arenotzero.
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– jizhou
Apr17,2016at4:54
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No,ofcoursethey'renotthesame.I'msorryforthedumbcomment.ThetermforwhatJohnRoeissayingis"animmersion".
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– user98602
Apr17,2016at4:56
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Therearetwodifferentdefinitionsinuse-seee.g.en.wikipedia.org/wiki/….
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– AnthonyCarapetis
Apr17,2016at6:12
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ThanksMikeandAnthonyforthecomments.Ithinkthedefinition"GivenadifferentiablemapffromRmintoRn,thecriticalpointsoffarethepointsofRm,wheretherankoftheJacobianmatrixoffisnotmaximal"(fromwikipointedoutbyAnthony)isbetterthanaskingrank(f)=m.Andinthebook"AnintroductiontoManifolds"itseemsthatthesurjectivepropertyisnotusedinanyproof.
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– jizhou
Apr17,2016at7:10
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