Some properties of normal sections and geodesics

I. I. Bodrenko

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2014), pp. 6-16.

Résumé

Let $F^{n}$ be $n$-dimensional $(n \geq 2)$ submanifold in $(n+p)$-dimensional Euclidean space $E^{n+p}$ $(p \geq 1)$. Let $x$ be arbitrary point $F^n$, $T_xF^n$ be tangent space to $F^n$ at the point $x$. Let $\gamma_g(x, t)$ be a geodesic on $F^n$ passing through the point $x\in F^n$ in the direction $t\in T_x F^n$. Denote by $k_g (x, t)$ and $\varkappa_g (x, t)$ curvature and torsion of geodesic $\gamma_g (x, t)\subset E^{n+p}$, respectively, calculated for point $x$. Torsion $\varkappa_g(x, t)$ of geodesic $\gamma_g (x, t)$ is called geodesic torsion of submanifold $F^n\subset E^{n+p}$ at the point $x$ in the direction $t$. Let $\gamma_N(x, t)$ be a normal section of submanifold $F^n\subset E^{n+p}$ at the point $x\in F^n$ in the direction $t\in T_xF^n$. Denote by $k_N (x, t)$ and $\varkappa_N (x, t)$ curvature and torsion of normal section $\gamma_N (x, t)\subset E^{n+p}$, respectively, calculated for point $x$. Denote by $b$ the second fundamental form of $F^n$, by $\overline\nabla$ the connection of van der Waerden — Bortolotti. The fundamental form $b\not=0$ is called cyclic recurrent if on $F^n$ there exists $1$-form $\mu$ such that $$ \overline\nabla_X b(Y,Z)= \mu(X)b(Y,Z) + \mu(Y)b(Z,X)+ \mu(Z)b(X,Y) $$ for all vector fields $X, Y, Z$ tangent to $F^n$. Submanifold $F^n\subset E^{n+p}$ with cyclic recurrent the second fundamental form $b\ne 0$ is called cyclic recurrent submanifold. The properties of normal sections $\gamma_N(x, t)$ and geodesics $\gamma_g(x, t)$ on cyclic recurrent submanifolds $F^n\subset E^{n+p}$ are studied in this article. The conditions for which cyclic recurrent submanifolds $F^n \subset E^{n+p}$ have zero geodesic torsion $\varkappa_g(x, t)\equiv 0$ at every point $x\in F^n$ in every direction $t\in T_xF^n$ are derived in this article. Denote by ${\mathcal R}_0$ a set of submanifolds $F^n\subset E^{n+p}$, on which $$ k_g (x,t)\ne 0, \quad \varkappa_g(x,t)\equiv 0, \quad \forall x\in F^n, \quad \forall t\in T_xF^n. $$ The following theorem is proved in this article. Let $F^n$ be a cyclic recurrent submanifold in $E^{n+p}$ with no asymptotic directions. Then $F^n$ belongs to the set ${\mathcal R}_0$ if and only if the following condition holds: $$ k_N(x, t) = k(x), \quad \forall x\in F^n, \quad \forall t\in T_xF^n. $$

Détail

Keywords: the second fundamental form, cyclic recurrent submanifold, geodesic torsion, normal curvature, connection of van der Waerden — Bortolotti.
Mots-clés : normal section, normal torsion

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