Differential Geometry
Homework 6

03.07.08
Hava Shabtai, ID 043039619,
Department of Mathematics, University of Haifa
Email: —hshabtai@campus.haifa.ac.il—
1. Let
φ
:
U
→
X
be the char described in Theorem 1.11
ϕ
(
u
) = (
u,f
(
u
)) = (
u
1
,...,u
n
,f
(
u
1
,...,u
n
)) =
= (
u
1
,...,u
n
,f
1
(
u
)
,...,f
m
(
u
))
.
This is a smooth parametrization of
X
, and the induced metric on
X
is given in
graph coordinates by
ϕ
*
g
=
ϕ
*
(
(
dx
1
)
2
+
···
+ (
dx
N
)
2
)
=
= (
du
1
)
2
+
···
+ (
du
n
)
2
+ (
df
1
)
2
+
···
+ (
df
m
)
2
2. The ﬁrst embedding is not isometry and the second embedding is an isometry.
(a) Let us take the geodesic line on
T
2
between the points
(0
,
0)
and
(0
,π/
2)
, its
length is
π/
2
as we can easily compute, however the end point of the image
of the geodesic line are
i
(0
,
0) = (0
,
0
,
0)
and
i
(0
,π/
2) = (0
,
2
,
0)
hence the
length of the image of the geodesic line is 2, since isometry must preserve the
distance between two points this map is not an isometry.
(b) Let us compute