Geodesic
Geometry of Position Function of Totally Real Submanifolds in Complex Euclidean Spaces
Kragujevac Journal of Mathematics, Tome 37 (2013) no. 2, p. 201 .

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A submanifold of a Euclidean space is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. The notion of constant-ratio submanifolds was first introduced and studied by the author in [5, 8] during the early 2000s. Such submanifolds relate to a problem in physics concerning the motion in a central force field which obeys the inverse-cube law of Newton (cf. [1, 15]). Recently, it was pointed out in [13] that constant-ratio submanifolds also relate closely to D'Arcy Thomson's basic principal of natural growth in biology. In this paper, we provide a fundamental study of totally real submanifolds of ${\bf C}^m$ in terms of the positive function $x$ of the submanifolds and the complex structure $J$ of ${\bf C}^m$. In particular, we classify constant-ratio totally real submanifolds in ${\bf C}^m$. Some related results are also obtained.
Classification : 53C40 53C42
Keywords: Totally real immersion, Spherical submanifold, Totally real cone, Constant ratio submanifold, Equiangular submanifold 
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     author = {Bang-Yen Chen},
     title = {Geometry of {Position} {Function} of {Totally} {Real} {Submanifolds} in {Complex} {Euclidean} {Spaces}},
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Bang-Yen Chen. Geometry of Position Function of Totally Real Submanifolds in Complex Euclidean Spaces. Kragujevac Journal of Mathematics, Tome 37 (2013) no. 2, p. 201 . http://geodesic.mathdoc.fr/item/KJM_2013_37_2_a0/