- #1
chartery
- 40
- 4
I'm having trouble with notations and visualisations regarding Ricci curvature.
For Riemann tensor there is variously:
##R^{\rho}\text{ }_{\sigma\mu\nu}\text{ }X^{\mu}Y^{\nu}V^{\sigma}\partial_{\rho}##
##[\nabla _{X},\nabla _{Y}]V##
##R(XY)V\mapsto Z##
##\left\langle R(XY)V,Z \right\rangle## i.e. covariant ##R(XYVZ)##
which I think I can grasp, mostly.
So my pictorial visualisation is of Z as the result of transporting vector V around loop given by vectors X,Y.
It even makes some sense then regarding it (Riem) as a sectional Gaussian Curvature of 2D subplane defined by X,Y.
But I run into sand with Ricci tensor being an average of these subplanes.
1) Does ##R^{\rho}\text{ }_{\sigma\rho\nu}\text{ }X^{\rho}Y^{\nu}V^{\sigma}\partial_{\rho}\text{ }## become ##R(XY)V\mapsto X## and ##\left\langle R(XY)V,X \right\rangle## ?
2) Does ##(R^{\rho}\text{ }_{\sigma\rho\nu}\text{ }X^{\rho}Y^{\nu}V^{\sigma}\partial_{\rho}\text{ }=)\text{ }R_{\sigma\nu}Y^{\nu}V^{\sigma}## become ##Ric(YV)## or ##Ric(XX)## or something else ?
3) Gaussian Curvature seems to be given as ##K(yv)## proportional to ##\left\langle R(yv)v,y \right\rangle## which has both duplicated, rather than just the ##x##, as in 1) above ?
(I have used ##yv## instead of usual ##uv## to try and keep track of the contraction.)
4) Is there a way to picture how contracting the Z and X vectors (leaving the V and Y) would lead to averaging of XY subplanes?
For Riemann tensor there is variously:
##R^{\rho}\text{ }_{\sigma\mu\nu}\text{ }X^{\mu}Y^{\nu}V^{\sigma}\partial_{\rho}##
##[\nabla _{X},\nabla _{Y}]V##
##R(XY)V\mapsto Z##
##\left\langle R(XY)V,Z \right\rangle## i.e. covariant ##R(XYVZ)##
which I think I can grasp, mostly.
So my pictorial visualisation is of Z as the result of transporting vector V around loop given by vectors X,Y.
It even makes some sense then regarding it (Riem) as a sectional Gaussian Curvature of 2D subplane defined by X,Y.
But I run into sand with Ricci tensor being an average of these subplanes.
1) Does ##R^{\rho}\text{ }_{\sigma\rho\nu}\text{ }X^{\rho}Y^{\nu}V^{\sigma}\partial_{\rho}\text{ }## become ##R(XY)V\mapsto X## and ##\left\langle R(XY)V,X \right\rangle## ?
2) Does ##(R^{\rho}\text{ }_{\sigma\rho\nu}\text{ }X^{\rho}Y^{\nu}V^{\sigma}\partial_{\rho}\text{ }=)\text{ }R_{\sigma\nu}Y^{\nu}V^{\sigma}## become ##Ric(YV)## or ##Ric(XX)## or something else ?
3) Gaussian Curvature seems to be given as ##K(yv)## proportional to ##\left\langle R(yv)v,y \right\rangle## which has both duplicated, rather than just the ##x##, as in 1) above ?
(I have used ##yv## instead of usual ##uv## to try and keep track of the contraction.)
4) Is there a way to picture how contracting the Z and X vectors (leaving the V and Y) would lead to averaging of XY subplanes?