Local differential geometry of singular curves with finite multiplicities. (English) Zbl 1390.53006

Fukui, Toshizumi

Saitama Math. J. 31, 79-88 (2017).

Given a \(C^{\infty}\) curve \(f\) in \(\mathbb R^n\) such that \(f(0)=0\), \(f\) is said to have multiplicity \(m\) at \(t=0\) if there exists a \(C^{\infty}\) map \(\tilde f\) such that \(f(t)=\frac{t^m}{m}\tilde f(t)\) with \(\tilde f(0)\neq 0\). The author’s first result is that if a curve has multiplicity \(m\) at \(0\) then there is a \(C^{\infty}\) parameter \(u\) such that \(\frac{u^m}{m}\) is an arc length

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parameter.
The second Theorem rediscovers a closed formula

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for the curvatures of a curve \(f\) of multiplicity \(m\) in terms of Gram determinants

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. Such a formula seems to have been known before by Gluck in the 60’s but without a complete proof so the author gives a rigurous proof of this formula. As a consequence of this result the author is able to prove the last main theorem which is a version of the fundamental theorem of curves at a singular point

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. More precisely, the author proves that given \(n-1\) \(C^{\infty}\) functions such that the first \(n-2\) ones are positive for \(t\neq 0\), then there exists a curve (unique up to motions in \(\mathbb R^n\)) of multiplicity \(m\) such that \(\frac{t^m}{m}\) is the arc length parameter and the curvatures of the curve are given by the \(n-1\) functions divided by \(t^{m-1}\).

Reviewer: Raúl Oset Sinha (València)

MSC:

53A04	Curves in Euclidean and related spaces
53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Keywords:

curves; multiplicity; arc length parameter; curvatures