dennis westra
Nov17-04, 06:38 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I have a basic question on the computation of the de Rahm cohomology\non a group manifold. What I need to know is how to choose the basis of\nclosed forms which are not exact.\n\n[Moderator\'s note: There are many different ways how can you choose\na basis of a linear space. Some bases are more convenient, but others\nare bases, too. ;-) LM]\n\nCan I take just the left-invariant forms and multiply them with constant\nnumbers, or should these constant also depend on the coordinates on the\ngroup manifold.\n\n[Moderator\'s note: I thought that if something depends on coordinates of\nyour manifold, then it\'s exactly what\'s called a "non-constant". ;-)\nAlso, if you multiply a closed form by a scalar function i.e. a 0-form\n(which is what you call a constant), the product will\nnot be closed, but will be proportional to the wedge product\nof the original closed form and the gradient of the 0-form. LM]\n\nIn other words; does it suffice to calculate the betti numbers for forms\n\\Phi_{a_1 ... a_p}\\sigma^a_1 \\wedge ...\\wedge \\sigma^a_p with \\Phi\nconstants and the \\sigma\'s left-invariant Maurer-Cartan forms? Since forms\nare allowed to depend on the coordinates I would think it differs from\ndoing the analysis with \\Phi non-constant... if I take the group manifold\nto be compact, can a general statement be made?\n\n[Moderator\'s note: Please, who has more time, answer this! Tx, LM]\n\nthanks for yoyr time,\n\\dw\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I have a basic question on the computation of the de Rahm cohomology
on a group manifold. What I need to know is how to choose the basis of
closed forms which are not exact.
[Moderator's note: There are many different ways how can you choose
a basis of a linear space. Some bases are more convenient, but others
are bases, too. ;-) LM]
Can I take just the left-invariant forms and multiply them with constant
numbers, or should these constant also depend on the coordinates on the
group manifold.
[Moderator's note: I thought that if something depends on coordinates of
your manifold, then it's exactly what's called a "non-constant". ;-)
Also, if you multiply a closed form by a scalar function i.e. a 0-form
(which is what you call a constant), the product will
not be closed, but will be proportional to the wedge product
of the original closed form and the gradient of the 0-form. LM]
In other words; does it suffice to calculate the betti numbers for forms
\Phi_{a_1 ... a_p}\sigma^a_1 \wedge ...\wedge \sigma^a_p with \Phi
constants and the \sigma's left-invariant Maurer-Cartan forms? Since forms
are allowed to depend on the coordinates I would think it differs from
doing the analysis with \Phi non-constant... if I take the group manifold
to be compact, can a general statement be made?
[Moderator's note: Please, who has more time, answer this! Tx, LM]
thanks for yoyr time,
\dw
on a group manifold. What I need to know is how to choose the basis of
closed forms which are not exact.
[Moderator's note: There are many different ways how can you choose
a basis of a linear space. Some bases are more convenient, but others
are bases, too. ;-) LM]
Can I take just the left-invariant forms and multiply them with constant
numbers, or should these constant also depend on the coordinates on the
group manifold.
[Moderator's note: I thought that if something depends on coordinates of
your manifold, then it's exactly what's called a "non-constant". ;-)
Also, if you multiply a closed form by a scalar function i.e. a 0-form
(which is what you call a constant), the product will
not be closed, but will be proportional to the wedge product
of the original closed form and the gradient of the 0-form. LM]
In other words; does it suffice to calculate the betti numbers for forms
\Phi_{a_1 ... a_p}\sigma^a_1 \wedge ...\wedge \sigma^a_p with \Phi
constants and the \sigma's left-invariant Maurer-Cartan forms? Since forms
are allowed to depend on the coordinates I would think it differs from
doing the analysis with \Phi non-constant... if I take the group manifold
to be compact, can a general statement be made?
[Moderator's note: Please, who has more time, answer this! Tx, LM]
thanks for yoyr time,
\dw