View source for Geodesic curvature

''at a point of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440701.png" /> on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440702.png" />''

The rate of rotation of the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440703.png" /> around the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440704.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440705.png" />, i.e. the projection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440706.png" /> of the vector of the angular rate of rotation of the tangent moving along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440707.png" />. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g0440709.png" /> are regular and oriented, and that the velocity is taken relative to the arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407010.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407011.png" />. The geodesic curvature can be defined as the curvature of the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407012.png" /> on the plane tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407013.png" /> at the point under consideration. The geodesic curvature is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407014.png" /></td> </tr></table>

where a prime denotes differentiation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407015.png" />.

The geodesic curvature forms part of the function expressing the variation of the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407016.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407017.png" /> is varied on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407018.png" />. If the ends are fixed:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407019.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407020.png" /> is the vector of variation of the curve. Curves for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407021.png" /> are geodesic lines (cf. [[Geodesic line|Geodesic line]]).

The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407022.png" /> is called the total geodesic curvature, or the rotation, of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407023.png" />. The connection between the rotation of a closed contour and the total curvature of the included region on the surface is given by the [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]].

The geodesic curvature forms a part of the [[Interior geometry|interior geometry]] of the surface, and can be expressed in terms of the metric tensor and the derivatives of the intrinsic surface coordinates with respect to its parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407024.png" />. If the geometry of a Riemannian space is studied without considering the latter to be immersed in Euclidean space, then the geodesic curvature is the only curvature which can be defined for a curve and the word  "geodesic"  is omitted. In considering curves on a submanifold of a Riemannian space, the curvature of a curve may be defined in the external space and in the submanifold — just like the spatial curvature and the geodesic curvature of a curve on a surface.

The concept of the geodesic curvature may be introduced for a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407025.png" /> on a general convex surface. If the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407026.png" /> has a length and each one of its arcs has a certain rotation, the right (left) geodesic curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407027.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407028.png" /> is the limit of the ratio of the right (left) rotation of the arc to its length, under the condition that the arc is contracted towards the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407029.png" />.

Two concepts of geodesic curvature are defined in a Finsler space. These differ in the manner in which the length of the vector replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044070/g04407030.png" /> is determined. On geodesics these geodesic curvatures are zero.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. O'Neill,   "Elementary differential geometry" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>