unicity of a 1-form
0
$begingroup$
I read somewhere that if the 1st cohomology group of a manifold is trivial, then it contains a 1-form that is unique up to an exact form.
Is this statement true?
Thank you!
differential-geometry
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
$endgroup$
add a comment |
0
$begingroup$
I read somewhere that if the 1st cohomology group of a manifold is trivial, then it contains a 1-form that is unique up to an exact form.
Is this statement true?
Thank you!
differential-geometry
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
$endgroup$
2
$begingroup$
If the first de Rham cohomology group is trivial, then every closed one-form is exact, but there are plenty of non-exact one-forms.
$endgroup$
– Michael Albanese
Dec 22 '18 at 0:20
$begingroup$
Are you sure? if the first Rham cohomology group is trivial, then every closed form is exact, there can be no non-exact 1-form, since the Rham cohomology group is the quotient of closed forms over exact forms. Could you please give details?
$endgroup$
– PerelMan
Dec 22 '18 at 0:46
1
$begingroup$
The point is that not every form is closed.
$endgroup$
– Aleksandar Milivojevic
Jan 5 at 20:49
add a comment |
0
0
0
$begingroup$
I read somewhere that if the 1st cohomology group of a manifold is trivial, then it contains a 1-form that is unique up to an exact form.
Is this statement true?
Thank you!
differential-geometry
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
$endgroup$
I read somewhere that if the 1st cohomology group of a manifold is trivial, then it contains a 1-form that is unique up to an exact form.
Is this statement true?
Thank you!
differential-geometry
differential-geometry
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
asked Dec 22 '18 at 0:03
PerelManPerelMan
760414
760414
2
$begingroup$
If the first de Rham cohomology group is trivial, then every closed one-form is exact, but there are plenty of non-exact one-forms.
$endgroup$
– Michael Albanese
Dec 22 '18 at 0:20
$begingroup$
Are you sure? if the first Rham cohomology group is trivial, then every closed form is exact, there can be no non-exact 1-form, since the Rham cohomology group is the quotient of closed forms over exact forms. Could you please give details?
$endgroup$
– PerelMan
Dec 22 '18 at 0:46
1
$begingroup$
The point is that not every form is closed.
$endgroup$
– Aleksandar Milivojevic
Jan 5 at 20:49
add a comment |
2
$begingroup$
If the first de Rham cohomology group is trivial, then every closed one-form is exact, but there are plenty of non-exact one-forms.
$endgroup$
– Michael Albanese
Dec 22 '18 at 0:20
$begingroup$
Are you sure? if the first Rham cohomology group is trivial, then every closed form is exact, there can be no non-exact 1-form, since the Rham cohomology group is the quotient of closed forms over exact forms. Could you please give details?
$endgroup$
– PerelMan
Dec 22 '18 at 0:46
1
$begingroup$
The point is that not every form is closed.
$endgroup$
– Aleksandar Milivojevic
Jan 5 at 20:49
2
2
$begingroup$
If the first de Rham cohomology group is trivial, then every closed one-form is exact, but there are plenty of non-exact one-forms.
$endgroup$
– Michael Albanese
Dec 22 '18 at 0:20
$begingroup$
If the first de Rham cohomology group is trivial, then every closed one-form is exact, but there are plenty of non-exact one-forms.
$endgroup$
– Michael Albanese
Dec 22 '18 at 0:20
$begingroup$
Are you sure? if the first Rham cohomology group is trivial, then every closed form is exact, there can be no non-exact 1-form, since the Rham cohomology group is the quotient of closed forms over exact forms. Could you please give details?
$endgroup$
– PerelMan
Dec 22 '18 at 0:46
$begingroup$
Are you sure? if the first Rham cohomology group is trivial, then every closed form is exact, there can be no non-exact 1-form, since the Rham cohomology group is the quotient of closed forms over exact forms. Could you please give details?
$endgroup$
– PerelMan
Dec 22 '18 at 0:46
1
1
$begingroup$
The point is that not every form is closed.
$endgroup$
– Aleksandar Milivojevic
Jan 5 at 20:49
$begingroup$
The point is that not every form is closed.
$endgroup$
– Aleksandar Milivojevic
Jan 5 at 20:49
add a comment |
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2
$begingroup$
If the first de Rham cohomology group is trivial, then every closed one-form is exact, but there are plenty of non-exact one-forms.
$endgroup$
– Michael Albanese
Dec 22 '18 at 0:20
$begingroup$
Are you sure? if the first Rham cohomology group is trivial, then every closed form is exact, there can be no non-exact 1-form, since the Rham cohomology group is the quotient of closed forms over exact forms. Could you please give details?
$endgroup$
– PerelMan
Dec 22 '18 at 0:46
1
$begingroup$
The point is that not every form is closed.
$endgroup$
– Aleksandar Milivojevic
Jan 5 at 20:49