The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.

中文翻译：

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在平滑表面上校正和闭合曲线

本文的主要目的是研究在光滑表面上校正和使曲线平滑的现象。在本文中，我们找到了平滑表面上矫正曲线的法向曲率和测地曲率，并且证明了在曲面等距情况下测地曲率是不变的，从而保留了矫正曲线。我们找到了一个充分条件，使得光滑曲面上的密合曲线在曲面等轴测条件下保持不变，并且我们证明了平滑曲面上的密合曲线α（s）的位置矢量沿着与曲面的任何切向量的分量在等轴测图上，α（s）处的不变。