%0 Journal Article %1 890.53003 %A Ivanova-Karatopraklieva, Ivanka %D 1993 %J Tensor, New Ser. %K bending; fields; fundamental higher infinitesimal order rigidity} rotational surfaces; {closed %P 1-9 %T On the infinitesimal bendings of higher order of closed rotational surfaces. %V 54 %X Let a radius vector of a closed rotational surface $S$ in $\bbfR\sp 3$ and its infinitesimal bending (inf. b.) $St$ of $m$-th order be given by $S:r(z,þeta) =ze3+\rho(z)e (þeta)$, $St:r(z,þeta,t) =r(z,þeta) +\summj=1 tjj z (z,þeta)$, where $z0,1$, $e(þeta)=e1þeta+ e2þeta$, $þeta0,2\pi$. Following S. E. Cohn-Vossen, let $øverset 1z (z,þeta)=1zk (z, þeta)$, $k2$, be a fundamental field of inf. b. of first order and $j\to z (z,þeta)$, $j=1, \dots, m$ be its extensions to fields of inf. b. to the order $j$. It is stated the following.Let $S$ have a regular $1zk$, where $k>A(m,n)= 1m n(m-1) \bigln(2m-1) -2m\bigr n-1$, and $m$ be even. Assume that $z1k$ can be extended to a regular field $m-1\to z$ of inf. b. of $(m-1)$-th order. If $S$ has no regular fundamental fields $1z2k$, $1z4k, \dots, 1zmk$, of inf. b. of first order, then the field $1zk$ can be extended to a regular field $øverset mz$ of inf. b. of $m$-th order; if $S$ has a regular fundamental field $1z2hk$, $1<2hm$, of inf. b. of 1-st order, then the field $1zk$ can be extended to a regular field $m z$ of inf. b. of $m$-th order if and only if a special condition is true.Analogous results are obtained for odd $m$. Some corollaries are derived regarding nonrigidity of $m$-th order of $S$.

@article{890.53003, abstract = {{Let a radius vector of a closed rotational surface $S$ in $\bbfR\sp 3$ and its infinitesimal bending (inf. b.) $S\sb t$ of $m$-th order be given by $S:r(z,\theta) =ze\sb 3+\rho(z)e (\theta)$, $S\sb t:r(z,\theta,t) =r(z,\theta) +\sum\sp m\sb{j=1} t\sp j{\overset j \to z} (z,\theta)$, where $z\in [0,1]$, $e(\theta)=e\sb 1\cos \theta+ e\sb 2\sin \theta$, $\theta\in [0,2\pi]$. Following S. E. Cohn-Vossen, let ${\overset 1\to z} (z,\theta)={\overset 1\to z}\sb k (z, \theta)$, $k\ge 2$, be a fundamental field of inf. b. of first order and ${\overset j\to z} (z,\theta)$, $j=1, \dots, m$ be its extensions to fields of inf. b. to the order $j$. It is stated the following.\par Let $S$ have a regular ${\overset 1\to z}\sb k$, where $k>A(m,n)= {1\over m} \sqrt{{n(m-1) \bigl[n(2m-1) -2m\bigr] \over n-1}}$, and $m$ be even. Assume that $z\sp 1\sb k$ can be extended to a regular field $\overset m-1\to z$ of inf. b. of $(m-1)$-th order. If $S$ has no regular fundamental fields ${\overset 1\to z}\sb{2k}$, ${\overset 1\to z}\sb{4k}, \dots, {\overset 1\to z}\sb{mk}$, of inf. b. of first order, then the field $\overset 1\to z\sb k$ can be extended to a regular field $\overset m\to z$ of inf. b. of $m$-th order; if $S$ has a regular fundamental field $\overset 1\to z\sb{2hk}$, $1<2h\le m$, of inf. b. of 1-st order, then the field $\overset 1\to z\sb k$ can be extended to a regular field $\overset m \to z$ of inf. b. of $m$-th order if and only if a special condition is true.\par Analogous results are obtained for odd $m$. Some corollaries are derived regarding nonrigidity of $m$-th order of $S$.} }, added-at = {2008-03-02T02:12:02.000+0100}, author = {Ivanova-Karatopraklieva, Ivanka}, biburl = {https://www.bibsonomy.org/bibtex/2c2b927e79167ec81da1fd8ac91557a33/dmartins}, classmath = {{*53A05 Surfaces in Euclidean space}}, description = {robotica-bib}, interhash = {8d1bcf24ec91c167efe8d0068d917402}, intrahash = {c2b927e79167ec81da1fd8ac91557a33}, journal = {Tensor, New Ser.}, keywords = {bending; fields; fundamental higher infinitesimal order rigidity} rotational surfaces; {closed}, language = {English}, pages = {1-9}, reviewer = {{Ue.Lumiste (Tartu)}}, timestamp = {2008-03-02T02:13:15.000+0100}, title = {{On the infinitesimal bendings of higher order of closed rotational surfaces.} }, volume = 54, year = 1993 }