Curvature on product Riemannian manifolds I am working on the following problem from Lee's Riemannian Manifolds: Suppose $g = g_1 \oplus g_2$ is a product metric on $M_1 \times M_2$   (i.e. $$g(X_1+X_2,Y_1+Y_2) = g_1(X_1,Y_1)+g_2(X_2,Y_2),$$ where $$X = X_1 + X_2, Y = Y_1+Y_2 \in T_{(p_1,p_2)}(M_1 \times M_2) = T_{p_1}M_1 \oplus T_{p_2}M_2.$$
(a) Show that for each point $p_i \in M_i$, the submanifolds $M_1 \times \{ p_2 \}$ and $\{ p_1 \} \times M_2$ are totally geodesic.
(b) If II $\subseteq T(M_1 \times M_2)$ is a 2-plane spanned by $X_1 \in TM_1$ and $X_2 \in TM_2$, show that $K$(II) $= 0$ (the sectional curvature is $0$).
(c) Show that the product metric on $S^2 \times S^2$ has nonnegative sectional curvature.
(d) Show that there is an embedding of $T^2$ in $S^2 \times S^2$ such that the induced metric is flat.
For part (a), I think I should be showing that the second fundamental form vanishes identically, but I'm not having any luck actually proving that.  I tried using the Weingarten equation, but got nowhere.  For part (b), I think that, using the formula for sectional curvature, this reduces to showing that $Rm(X_1,X_2,X_2,X_1) = 0$, but again I'm not having any luck proving that.  For (c), I think that if we take an arbitrary plane and use an orthonormal basis, we might be able to get something, but to be honest I'm just totally lost.  I have no idea for part (d).  Any help is MUCH appreciated!!!

Recall that Riemannian distance is infimum of length of curves (Here
length is calculated by using Riemannian metric) Hence product has a
product metric $g:=g_1+ g_2$ so that we have formula $$
d((x_1,y_1),(x_2,y_2)) = \sqrt{d_1(x_1,y_1)^2 + d_2(x_2,y_2)^2} $$
(a) Note that distance between two points $(x_i,p_2)$ in $M_1\times
\{p_2\}$ is equal to distance between $x_i$. Hence we concluded that
a geodesic curve in $M_1$ is in fact a geodesic curve in $M_1\times
p_2$ Hence $M_1$ is a totally geodesic
(b) $E_i,\ E_\alpha$ are coordinate vector fields in $M_i$
respectively $$ \Gamma_{\alpha \beta}^j = \frac{1}{2} g^{jk} (g_{k
\alpha,\beta } +g_{k\beta,\alpha}-g_{\alpha\beta,k} ) =0
$$
By calculating others, ${\rm sec}\
(E_i,E_\alpha)=0$
(c) If $M_i=S^2$, then $$ {\rm sec}\ (E_i,E_j)={\rm sec}\ (S^2),\
{\rm sec}\ (E_i,E_\alpha ) =0 $$
(d) $T^2=S^1\times S^1\subset S^2\times S^2$ Here $S^1$ is a
geodesic curve in $S^2$. Then $T^2$ is totally geodesic so that $
{\rm sec}\ (T^2)=0$
