TITLE: generalized warped product
QUESTION [1 upvotes]: Let $(M_1,g_1)$, $(M_2,g_2)$ denote two Riemannian manifolds, let $(I,dt^2)$ be the unit interval with its standard metric.
I would like to study the manifold $(M,g)$ where $M = I \times (M_1 \times M_2)$ and $g = dt^2 + f_1^2(t)g_1 + f^2_2(t)g_2$, (i.e. each metric component is multiplied by a positive factor $f^2_i > 0$ depending only on $t$. 
I was wondering whether there a good reference for this kind of problem, so far I have found a lot of material on warped products (which looks similar) but had little success with regards to the above setting. 
Many thanks for your help!

REPLY [1 votes]: use this one http://www.sciencedirect.com/science/article/pii/S0393044099000728
Also this one is good
http://arxiv.org/pdf/math/0406039.pdf definition 2.1 and the same article in also in sciencedirect
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