Math 497C
Nov 30, 2004
1
Curves and Surfaces
Fall 2004, PSU
Lecture Notes 15
2.13
The Geodesic Curvature
Let
α
:
I
→
M
be a unit speed curve lying on a surface
M
⊂
R
3
. Then the
absolute geodesic curvature
of
α
is defined as

κ
g

:=
(
α
)
=
α

α , n
(
α
)
n
(
α
)
,
where
n
is a local Gauss map of
M
in a neighborhood of
α
(
t
). In particular
note that if
M
=
R
2
, then

κ
g

=
κ
, i.e., absolute geodesic curvature of a
curve on a surface is a gneralization of the curvature of curves in the plane.
Exercise 1.
Show that the absolute geodesic curvature of great circles in a
sphere and helices on a cylinder are everywhere zero.
Similarly, the
(signed) geodesic curvature
generalizes the notion of the signed
curvature of planar curves and may be defined as follows.
We say that a surface
M
⊂
R
3
is
orientable
provided that there exists a
(global) Gauss map
n
:
M
→
S
2
, i.e., a
continuous
mapping which satisfies
n
(
p
)
∈
T
p
M
, for all
p
∈
M
. Note that if
n
is a global Gauss map, then so is

n
. In particular, any orientable surface admits precisely two choices for its
global Gauss map. Once we choose a Gauss map
n
for an orientable surface,
then
M
is said to be
oriented
.
If
M
is an oriented surface (with global Gauss map
n
), then, for every
p
∈
M
, we define a mapping
J
:
T
p
M
→
T
p
M
by
JV
:=
n
×
V.
Exercise 2.
Show that if
M
=
R
2
, and
n
= (0
,
0
,
1), then
J
is clockwise
rotation about the origin by
π/
2.
1
Last revised: December 6, 2007
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Then the
geodesic curvature
of a unit speed curve
α
:
I
→
M
is given by
κ
g
:=
α , Jα
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Math, Geometry, Manifold, Orientability, Riemannian geometry, Geodesic

Click to edit the document details