3] be an isometric immersion of a surface S in the Euclidean 3-space.

Oral: Bertrand Partner D-Curves in Euclidean 3-space E3, arXiv:1003.

The proper Euclidean 3-space [SIGMA]' and its underlying one-dimensional proper intrinsic space [phi][rho]' shall sometimes be referred to as our proper (or classical) Euclidean 3-space and our proper (or classical) intrinsic space for brevity.

There are likewise the proper Euclidean 3-space -[[SIGMA].

0,*] exist naturally, then they should belong to a new pair of worlds (or universes), just as the horizontal proper Euclidean 3-space [SIGMA]' and -[SIGMA]'* and their underlying one-dimensional isotropic proper intrinsic spaces [phi][rho]' and -[phi][rho]'* exist naturally and belong to the positive (or our) universe and the negative universe respectively, as found in [1] and [2].

Corresponding to every given point P in our proper Euclidean 3-space [SIGMA]', there are unique symmetry- partner point [P.

The one-dimensional intrinsic masses of all particles and objects are aligned along the singular isotropic one-dimensional intrinsic space [phi][rho], whose inertial masses are scattered arbitrarily in the physical Euclidean 3-space [SIGMA] with respect to these observers, in (E, ct), as illustrated for three such particles and objects in Fig.

The intrinsic mass [phi]m of a particle or object in the intrinsic space [phi][SIGMA] lies directly underneath the inertial mass m of the particle or object in the physical Euclidean 3-space [SIGMA], as illustrated for three such particles or objects in Fig.

The flat four-dimensional physical spacetime ([SIGMA], ct) containing the three-dimensional inertial masses m of particles and objects in the Euclidean 3-space [SIGMA] is the outward manifestation of the flat four-dimensional intrinsic spacetime ([phi][SIGMA], [phi]c[phi]t) containing the three-dimensional intrinsic masses [phi]m of the particles and objects in [phi][SIGMA] in Fig.

2], an interesting geometric question is raised: Classify all surfaces in Euclidean 3-space and Minkowski 3-space satisfying the conditions

In this paper, we would like to contribute the solution of the above question, by studying this question for tubes or tubular surfaces in Euclidean 3-space [E.

Tubular surfaces of Weingarten type in Euclidean 3-space

1a (to which the proper Euclidean 3-space [[SIGMA].

0]' along the vertical projects zero component (or nothing) into the Euclidean 3-space [SIGMA]' (as a hyper-surface) along the horizontal.

where [phi][rho]' is without the superscript "0" label because it lies in (or underneath) our Euclidean 3-space [SIGMA]' (without superscript "0" label) along the horizontal.