Frederic P Schuller
Oct22-04, 12:01 PM
<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>\n\nHi,\n\nLet (M,g) be a (pseudo-)Riemannian manifold, and \\Lambda^2 the space\nof antisymmetric tensors of second rank on M. On \\Lambda^2 there is a\nmetric G induced by\nG(X,Y,A,B) = g(X,A) g(Y,B) - g(X,B) g(Y,A)\nas G shares the Riemann-Christoffel symmetries\nG(X,Y,A,B) = -G(Y,X,A,B) = -G(X,Y,B,A)\nG(X,Y,A,B) = G(A,B,X,Y)\nThe geometrical significance of G is that G(X,Y,X,Y) measures the area\nsquared of the parallelogram defined by X and Y. Knowing only G, one\napparently cannot reconstruct the metric g.\n\nQuestion: Are there curvature invariants of (M,g) that can be written\nin terms of only G and its first and second derivatives, i.e. without\nusing the metric g?\n\nFrederic\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>Hi,
Let (M,g) be a (pseudo-)Riemannian manifold, and \Lambda^2 the space
of antisymmetric tensors of second rank on M. On \Lambda^2 there is a
metric G induced by
G(X,Y,A,B) = g(X,A) g(Y,B) - g(X,B) g(Y,A)
as G shares the Riemann-Christoffel symmetries
G(X,Y,A,B) = -G(Y,X,A,B) = -G(X,Y,B,A)G(X,Y,A,B) = G(A,B,X,Y)
The geometrical significance of G is that G(X,Y,X,Y) measures the area
squared of the parallelogram defined by X and Y. Knowing only G, one
apparently cannot reconstruct the metric g.
Question: Are there curvature invariants of (M,g) that can be written
in terms of only G and its first and second derivatives, i.e. without
using the metric g?
Frederic
Let (M,g) be a (pseudo-)Riemannian manifold, and \Lambda^2 the space
of antisymmetric tensors of second rank on M. On \Lambda^2 there is a
metric G induced by
G(X,Y,A,B) = g(X,A) g(Y,B) - g(X,B) g(Y,A)
as G shares the Riemann-Christoffel symmetries
G(X,Y,A,B) = -G(Y,X,A,B) = -G(X,Y,B,A)G(X,Y,A,B) = G(A,B,X,Y)
The geometrical significance of G is that G(X,Y,X,Y) measures the area
squared of the parallelogram defined by X and Y. Knowing only G, one
apparently cannot reconstruct the metric g.
Question: Are there curvature invariants of (M,g) that can be written
in terms of only G and its first and second derivatives, i.e. without
using the metric g?
Frederic