- #1
Sajet
- 48
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Hi!
I'm reading this script* and I fail to understand a rather simple calculation. I assume the problem lies in me not understanding the notation that is used, and I was unable to figure it out or find it in literature.
We have a smooth family of metrics [itex]g = g_t[/itex] on a Riemannian manifold, and we set [itex]h := \frac{\partial}{\partial t}g_t[/itex].
First question:
[itex]\frac{\partial}{\partial t} \nabla_X Y[/itex]: Does this mean [itex]\frac{\partial}{\partial t} \nabla_X^t Y[/itex], where [itex]\nabla^t[/itex] is the Levi-Civita connection w.r.t [itex]g_t[/itex]?
Second question:
The script says:
[itex]\langle \frac{\partial}{\partial t} \nabla_X Y, Z\rangle = \frac{\partial}{\partial t}g(\nabla_X Y, Z\rangle - h(\nabla_X Y, Z)[/itex]
I don't understand this step. Also I don't see the difference between the two terms
[itex]\frac{\partial}{\partial t}g(\nabla_X Y, Z\rangle[/itex] and
[itex]h(\nabla_X Y, Z)[/itex]
In class we defined
[itex]\frac{\partial g}{\partial t}(X, Y) := \frac{\partial}{\partial t}(g_t(X, Y))[/itex].
Therefore those two terms seem the same to me.
I would appreciate any help :)
* http://homepages.warwick.ac.uk/~maseq/RFnotes.html , p. 32.
I'm reading this script* and I fail to understand a rather simple calculation. I assume the problem lies in me not understanding the notation that is used, and I was unable to figure it out or find it in literature.
We have a smooth family of metrics [itex]g = g_t[/itex] on a Riemannian manifold, and we set [itex]h := \frac{\partial}{\partial t}g_t[/itex].
First question:
[itex]\frac{\partial}{\partial t} \nabla_X Y[/itex]: Does this mean [itex]\frac{\partial}{\partial t} \nabla_X^t Y[/itex], where [itex]\nabla^t[/itex] is the Levi-Civita connection w.r.t [itex]g_t[/itex]?
Second question:
The script says:
[itex]\langle \frac{\partial}{\partial t} \nabla_X Y, Z\rangle = \frac{\partial}{\partial t}g(\nabla_X Y, Z\rangle - h(\nabla_X Y, Z)[/itex]
I don't understand this step. Also I don't see the difference between the two terms
[itex]\frac{\partial}{\partial t}g(\nabla_X Y, Z\rangle[/itex] and
[itex]h(\nabla_X Y, Z)[/itex]
In class we defined
[itex]\frac{\partial g}{\partial t}(X, Y) := \frac{\partial}{\partial t}(g_t(X, Y))[/itex].
Therefore those two terms seem the same to me.
I would appreciate any help :)
* http://homepages.warwick.ac.uk/~maseq/RFnotes.html , p. 32.