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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.

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# D/dt g(t) and d/dt D_X Y

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