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Bibliography
Besse A. L.,Manifolds All of Whose Geodesics Are Closed, Springer, Berlin-Heidelberg-New York (1978).
Berger M. S. -Gauduchon P. -Mazet E.,Le spectre d'une variété Riemannienne, Lecture Notes in Math.,194, Springer (1971).
Bott R.,On manifolds all of whose geodesics are closed, Ann. Math.,60, 3 (1954), pp. 375–382.
Calabi E.,Minimal immersions of surfaces in Euclidean spheres, J. Diff. Geometry,1 (1967), pp. 111–125.
Chen B.-Y.,Geometry of Submanifolds, Marcel Dekker, New York (1973).
Chen B.-Y.,Geometry of Submanifolds and its Applications, Sci. Univ. Tokyo Press, Tokyo (1981).
Chen B.-Y.,Total Mean Curvature and Submanifolds of Finite Type, Worlds Scientific Publishing Co., New Jersey-Singapore (1984).
Chen B.-Y.,Submanifolds with planar normal sections, Soochow J. Math.,7 (1981), pp. 19–24.
Chen B.-Y.,Differential geometry of submanifolds with planat normal sections, Ann. Mat. Pura Appl.,130 (1982), pp. 59–66.
Chen B.-Y.,Classification of surfaces with planar normal sections, J. Geometry,20 (1983), pp. 122–127.
Chen B.-Y. -Verheyen P.,Sous-variétes dont les sections normales sont des géodésique, C. R. Acad. Sci. Paris, Sér. A,293 (1981), pp. 611–613.
Chen B.-Y. -Verheyen P.,Submanifolds with geodesic normal sections, Math. Ann.,269 (1984), pp. 417–429.
Chen B.-Y. -Deprez J. -Verheyen P.,Immersions, dans un espace euclidian, d'un espace symmetrique compact de rang un à géodésiques simples, C. R. Acad. Sci. Paris, Sér. I,304 (1987), pp. 567–570.
Chern S. S.,Minimal submanifolds in a Riemannian manifold, University of Kansas, Dept. of Math. Technical Report 19 (New Series), 1968.
Chern S. S. -do Carmo M. P. -Kobayashi S.,Minimal submanifolds of a sphere with second fundamental form of constant length, inFunctional Analysis and Related Fields, Springer, Berlin (1971), pp. 43–62.
Do Carmo M. P. -Wallach N. R.,Minimal immersions of spheres into spheres, Ann. Math.,95 (1971), pp. 43–62.
Erbacher J.,Reduction of codimension of isometric immersion, J. Diff. Geometry,5 (1971), pp. 333–340.
Ferus D.,Symmetric submanifolds of Euclidean space, Math. Ann.,247 (1980), pp. 81–93.
Ferus D. -Schirrmacher S.,Submanifolds in a Euclidean space with simple geodesics, Math. Ann.,260 (1982), pp. 57–62.
Green L. W.,Auf widersehenflächen, Ann. Math.,78 (1963), pp. 289–299.
Helgason S.,Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press (1978).
Hong S. L.,Isometric immersions of manifolds with plane geodesics into Euclidean space, J. Diff. Geometry,8 (1973), pp. 259–278.
Hong Y.,Helical submanifolds in Euclidean spaces, Indiana Univ. Math.,35 (1986), pp. 29–43.
Hong Y. -Houh C. S.,Helical immersions and normal sections, Kodai Math. J.,8 (1985), pp. 171–192.
Houh C. S. -Wang G.-Q.,Isotropic submanifolds with pointwise planar normal sections, J. Geometry,26 (1986), pp. 99–104.
Itoh T. -Ogiue K.,Isotropic immersions, J. Diff. Geometry,8 (1973), pp. 305–316.
Kobayashi S. -Nomizu K.,Foundations of Differential Geometry, Vol. I and II, Interscience, New York (1963, 1969).
Little J. A.,Manifolds with planar geodesics, J. Diff. Geometry,11 (1976), pp. 265–285.
Nakagawa H.,On certain minimal immersions of a Riemannian manifold into a sphere, Kodai Math. J.,3 (1980), pp. 321–340.
Ogiue K.,On Kaehler immersions, Canad. J. Math.,24 (1972), pp. 1178–1182.
O'Neill B.,Isotropic and Kaehler immersions, Canad. J. Math.,17 (1965), pp. 907–915.
Sakamoto K.,Planar geodesic immersions, Tôhoku Math. J.,29 (1977), pp. 25–56.
Sakamoto K.,Helical immersions into a unit sphere, Math. Ann.,261 (1982), pp. 63–80.
Sakamoto K.,On a minimal helical immersion into a unit sphere, inGeometry of Geodesics and Related Topics (Tokyo, 1982), Ad. Stud. Pure Math.,3, North-Holland, Amsterdam, New York (1984), pp. 193–211.
Sakamoto K.,Helical immersions of compact Riemannian manifolds into a unit sphere, Trans. Amer. Math. Soc.,288 (1985), pp. 765–790.
Sakamoto K.,Helical minimal imbeddings of order 4 into spheres, J. Math. Soc. Japan,37 (1985), pp. 315–336.
Sakamoto K.,Helical immersions into a Euclidean space, Michigan Math. J.,33 (1986), pp. 353–364.
Samelson H.,On manifolds with many closed geodesics, Portugaliae Mathematicae,22 (1963), pp. 193–196.
Tai S. S.,Minimum imbeddings of compact symmetric spaces of rank one, J. Diff. Geometry,2 (1968), pp. 55–66.
Takahashi T.,Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan,18 (1966), pp. 380–385.
Tsukada K.,Helical geodesic immersions of rank one symmetric spaces into spheres, Tokyo J. Math.,6 (1983), pp. 267–285.
Verheyen P.,Submanifolds with geodesic normal sections are helical, Rend. Sem. Mat. Politec. Torino,43 (1985), pp. 511–527.
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Kim, Y.H. Surfaces of a Euclidean space with Helical or planar geodesics through a point. Annali di Matematica pura ed applicata 164, 1–35 (1993). https://doi.org/10.1007/BF01759312
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DOI: https://doi.org/10.1007/BF01759312