Let $M_1$ and $M_2$ be Riemannian manifolds and consider the cartesian product $M_1 \times M_2$ with the profuct structure .Let $\pi_1: M_1 \times M_2 \to M_1$ and $\pi_2: M_1 \times M_2 \to M_2$be the natural projections.Introduce on $\ M_1 \times M_2$ a Riemannian metric as follows:
$$\langle u,v \rangle_{(p,q)} = \langle {dπ_1}_* u,{dπ_1}_* v \rangle_p + \langle {dπ_2}_*u,{dπ_2}_* v \rangle_q$$ .

How can I verify that this is actually a Riemannian metric.
To show this first I need to show that this bilinear and smooth. Am I right?
If yes then
(1) since derivative map is linear and inner product is bilinear do this is linear.is it ok?
(2) but how can I show that this is smooth?

Need your help.Please explain in detail.I am totally new in the subject.

  • $\begingroup$ Why it is negatively voted? Is there any problem? Should I get some help? $\endgroup$ – rahul Apr 14 '15 at 2:49
  • 2
    $\begingroup$ A Riemannian metric is first of all a tensor field, so you need to show that it is $C^\infty(M\times N, R)$-bilinear. Is this bilinear, and can you factor out smooth functions? This is enough to take care of smoothness (see John Lee's text). But a Riemannian metric has another requirement: It needs to be positive definite. Is $\langle X, X \rangle > 0$ for $X$ any non-zero vector field? $\endgroup$ – ಠ_ಠ Apr 14 '15 at 8:43

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