Page 344 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 344
DIFFERENTIAL GEOMETRY: OLD AND NEW (MS-15)
Characterization of manifolds of constant curvature by spherical curves
and ruled surfaces
Luiz C. B. Da Silva, luiz.da-silva@weizmann.ac.il
Weizmann Institute of Science, Israel
Coauthor: José D. Da Silva
Space forms, i.e., Riemannian manifolds of constant sectional curvature, play a prominent role
in geometry and an important problem consists of finding properties that characterize them.
In this talk, we report results from [1], where we show that the validity of some theorems
concerning curves and surfaces can be used for this purpose. For example, it is known that the
so-called rotation minimizing (RM) frames allow for a characterization of geodesic spherical
curves in Euclidean, hyperbolic, and spherical spaces through a linear equation involving the
coefficients that dictate the RM frame motion [2]. Here, we shall prove the converse, i.e., if
all geodesic spherical curves on a manifold are characterized by a certain linear equation, then
all the geodesic spheres with a sufficiently small radius are totally umbilical, and consequently,
the ambient manifold is a space form. (We also present an alternative proof, in terms of RM
frames, for space forms as the only manifolds where all geodesic spheres are totally umbilical
[3].) In addition, we furnish two other characterizations in terms of (i) an inequality involving
the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the
vanishing of the total torsion of closed spherical curves in the case of 3d manifolds. (These are
the converse of previous results [4].) Finally, we introduce ruled surfaces and show that if all
extrinsically flat surfaces in a 3d manifold are ruled, then the manifold is a space form.
References
[1] Da Silva, L.C.B. and Da Silva, J.D.: “Characterization of manifolds of constant curva-
ture by spherical curves". Annali di Matematica 199, 217 (2020); Da Silva, L.C.B. and
Da Silva, J.D.: “Ruled and extrinsically flat surfaces in three-dimensional manifolds of
constant curvature". Unpublished manuscript 2021.
[2] Bishop, R.L.: “There is more than one way to frame a curve". Am. Math. Mon. 82, 246
(1975); Da Silva, L.C.B. and Da Silva, J.D.: “Characterization of curves that lie on a
geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere".
Mediterr. J. Math. 15, 70 (2018)
[3] Kulkarni, R.S.: “A finite version of Schur’s theorem". Proc. Am. Math. Soc. 53, 440
(1975); Vanhecke, L. and Willmore, T.J.: “Jacobi fields and geodesic spheres". Proc. R.
Soc. Edinb. A 82, 233 (1979); Chen, B.Y. and Vanhecke, L.: “Differential geometry of
geodesic spheres". J. Reine Angew. Math. 325, 28 (1981).
[4] Baek, J., Kim, D.S., and Kim, Y.H.: “A characterization of the unit sphere". Am. Math.
Mon. 110, 830 (2003); Pansonato, C.C. and Costa, S.I.R., “Total torsion of curves in
three-dimensional manifolds". Geom. Dedicata 136, 111 (2008).
342
Characterization of manifolds of constant curvature by spherical curves
and ruled surfaces
Luiz C. B. Da Silva, luiz.da-silva@weizmann.ac.il
Weizmann Institute of Science, Israel
Coauthor: José D. Da Silva
Space forms, i.e., Riemannian manifolds of constant sectional curvature, play a prominent role
in geometry and an important problem consists of finding properties that characterize them.
In this talk, we report results from [1], where we show that the validity of some theorems
concerning curves and surfaces can be used for this purpose. For example, it is known that the
so-called rotation minimizing (RM) frames allow for a characterization of geodesic spherical
curves in Euclidean, hyperbolic, and spherical spaces through a linear equation involving the
coefficients that dictate the RM frame motion [2]. Here, we shall prove the converse, i.e., if
all geodesic spherical curves on a manifold are characterized by a certain linear equation, then
all the geodesic spheres with a sufficiently small radius are totally umbilical, and consequently,
the ambient manifold is a space form. (We also present an alternative proof, in terms of RM
frames, for space forms as the only manifolds where all geodesic spheres are totally umbilical
[3].) In addition, we furnish two other characterizations in terms of (i) an inequality involving
the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the
vanishing of the total torsion of closed spherical curves in the case of 3d manifolds. (These are
the converse of previous results [4].) Finally, we introduce ruled surfaces and show that if all
extrinsically flat surfaces in a 3d manifold are ruled, then the manifold is a space form.
References
[1] Da Silva, L.C.B. and Da Silva, J.D.: “Characterization of manifolds of constant curva-
ture by spherical curves". Annali di Matematica 199, 217 (2020); Da Silva, L.C.B. and
Da Silva, J.D.: “Ruled and extrinsically flat surfaces in three-dimensional manifolds of
constant curvature". Unpublished manuscript 2021.
[2] Bishop, R.L.: “There is more than one way to frame a curve". Am. Math. Mon. 82, 246
(1975); Da Silva, L.C.B. and Da Silva, J.D.: “Characterization of curves that lie on a
geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere".
Mediterr. J. Math. 15, 70 (2018)
[3] Kulkarni, R.S.: “A finite version of Schur’s theorem". Proc. Am. Math. Soc. 53, 440
(1975); Vanhecke, L. and Willmore, T.J.: “Jacobi fields and geodesic spheres". Proc. R.
Soc. Edinb. A 82, 233 (1979); Chen, B.Y. and Vanhecke, L.: “Differential geometry of
geodesic spheres". J. Reine Angew. Math. 325, 28 (1981).
[4] Baek, J., Kim, D.S., and Kim, Y.H.: “A characterization of the unit sphere". Am. Math.
Mon. 110, 830 (2003); Pansonato, C.C. and Costa, S.I.R., “Total torsion of curves in
three-dimensional manifolds". Geom. Dedicata 136, 111 (2008).
342
