Blaga P. Lectures on the differential geometry of - tiera.ru

4.19. Geodesics 177
will denote it by kg and we will call it geodesic or tangential curvature of the curve 7 . To
find the quantity d, we start from the relation
N = sin θ · ν − cos θ · β.
By differentiating with respect to the arclength of the curve, we get, using the last two
of the Frenet’s formulae:
N ′ = θ ′ · sin θ · ν − sin θ(−k · τ + χ · β) + θ ′ sin θ · β + χ · cos θ · ν =
= −k · sin θ · τ + (θ ′ + χ) · (cos θ · ν + sin θ · β) =
= −k · sin θ · τ + (θ ′ + χ) · n,
and, comparing with the second equation from (4.19.2), we get
d = θ ′ + χ.
This function is denoted by χg and it s called the geodesic torsion. His geometric meaning
will be made clear later. Clearly, we cannot claim, as we did with the geodesic
curvature, that the geodesic torsion is the torsion of the projection of the curve on the
tangent plane, as the torsion of that curve is always zero, while the geodesic torsion of
the given curve is not, usually.
The geodesic curvature plays a much more important role in the differential geometry
of surfaces than the geodesic torsion does, so we start by focusing on it. We notice, to
begin with, that
kg = τ ′ · N = −τ · N ′ . (4.19.4)
Since, as we saw earlier, N = n × τ, one obtains for kg the expression
i.e.
kg = τ ′ · N = τ ′ · (n × τ),
kg = (τ, τ ′ , n). (4.19.5)
This formula holds for naturally parameterized curves. Let us consider, now, an arbitrary
regular parameterized curve on S , given by the local equations u = u(t), v = v(t). We
have, therefore, r = r(u(t), v(t)). If we denote by a dot the differentiation with respect to
the parameter t along the curve, we get
τ ≡ dr
ds
dr dt 1
= = · ˙r,
dt ds ˙s
7 Here the term “tangential” refers to the tangent plane of the surface, not to the tangent line of the curve.
In fact, it can be shown that the geodesic curvature at a point of a curve lying on a surface is the signed
curvature of the projection of the curve on the tangent plane to the surface at that particular point.