Geometry of Position Function of Totally Real Submanifolds in Complex Euclidean Spaces

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 C in terms of the positive function x of the submanifolds and the complex structure J of C. In particular, we classify constant-ratio totally real submanifolds in C. Some related results are also obtained.