Differential Geometry
Homework 8

21.07.08
Hava Shabtai, ID 043039619,
Department of Mathematics, University of Haifa
Email: —[email protected]—
1. In order to prove the theorem we must use
Gauss’s Theorema Egregium
Let
X
⊂
R
3
be a 2dimensional submanifold and
g
the induced metric on
X
. For
any
x
∈
X
and any basis
{
e
1
, e
2
}
for
T
x
(
X
)
, the Gaussian curvature of
X
at
x
is
given by
K
=
R
(
e
1
, e
2
, e
2
, e
2
)

e
1

2

e
2

2
 h
e
1
, e
2
i
2
.
Now let us prove that if
dimX
= 2
for any arbitrary vector field
s, t, v, w
one has
R
(
s, t, v, w
) =
K
(
h
s, v
i h
t, w
i  h
s, w
i h
t, v
i
)
.
Since both sides of this equation are tensors, we can compute them in terms of any
basis. Let
{
e
1
, e
2
}
be any orthonormal basis for
T
x
(
X
)
, and define a shorter symbole
for the following components
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 Spring '09
 MICHAELPOLYAK
 Geometry, constant sectional curvature, arbitrary vector field, Theorema Egregium Let, Hava Shabtai

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