The following is the Exercise 1) a), Chapter 6 from do Carmo, Riemannian Geometry:
- Let $M_1$ and $M_2$ be Riemannian manifolds and consider $M_1 \times M_2$ with the product metric. Let $\nabla^1$ and $\nabla^2$ be the Riemannian connection in $M^1$ and $M^2$, respectively.
a) Show the Riemannian connection $\nabla$ of $M_1 \times M_2$ is given by $$\nabla_{X_1 + X_2}(Y_1 + Y_2) = \nabla^1_{X_1}Y_1 + \nabla^2_{X_2}Y_2,$$ where $X_1,Y_1 \in \mathcal{X}(M_1)$ and $X_2,Y_2 \in \mathcal{X}(M_2)$.
Here are my doubts:
Doubt 1) I'm trying to understand what exactly means, for example, the sum $X_1 + X_2$. Is it just a notation for $(X_1, X_2) \in \mathcal{X}(M_1) \times \mathcal{X}(M_2)$?
Doubt 2) Given $p = (p_1,p_2) \in M_1 \times M_2$, it's really commum to identify $$T_{p}(M_1 \times M_2) \equiv T_{p_1}M_1 \oplus T_{p_2}M_2.$$ Again, the meaning of the last direct sum is just $T_{p_1}M_1 \times T_{p_2}M_2$ ? In this case, the identification would be $$L : T_{p}(M_1 \times M_2) \rightarrow T_{p_1}M_1 \times T_{p_2}M_2$$ defined by $L(w) = (\alpha_1'(0), \alpha_2'(0)),$ where $w \in T_{p}(M_1\times M_2)$ is given by $w = c'(0)$, for $c(t) = (\alpha_1(t), \alpha_2(t))$ with $\alpha_1 : I \rightarrow M_1$ and $\alpha_2 : I \rightarrow M_2$? In this case, using the natural structure for the product $M_1\times M_2$ and the map $L$, I conclude that a basis for $T_{p_1}M_1 \times T_{p_2}M_2$ would be: $$\{\partial_1, ...,\partial_{n}, \partial_{n+1}, ..., \partial_{n+m}\},$$ where $\partial_i = (\partial^1_i,0)$, for $i = 1, ..., n$ and $\partial_i = (0, \partial^2_i)$, for $i = n+1, ..., n+m$ [here $\partial^1_i$ and $\partial^2_i$ are tangent vector of the basis of $T_{p_1}M_1$ and $T_{p_2} M_2$ respectively].
Doubt 3) In item a) above, I believe the author is using the identification "a tangent fild in $\mathcal{X} (M_1\times M_2)$ is given by $(X_1, X_2).$ In this case, what would mean $(X_1, X_2)(f),$ for $f \in C^{\infty}(M_1 \times M_2)$? What would be its expression on a local coordinate system of $M_1 \times M_2$ ? (I tried this using the basis I wrote in Doubt 2.)
Doubt 4) What's the exactly relation between $\mathcal{X}(M_1\times M_2)$ and $\mathcal{X}(M_1) \times \mathcal{X}(M_2)$ ?
What a tried for item a):
In general, given a Riemannian manifold $M$ of dimension $n$, it's Levi-Civita connection is given by: $$\nabla_{X}Y = \sum_{k=1}^n(X(b_k) + \sum_{i,j=1}^n a_{i} b_{j} \Gamma_{ij}^k)\partial_k,$$ where, in coordenates, $X = \sum_{i=1}^n a_i \partial_i$ and $Y = \sum_{j=1}^n b_j \partial_j$. I tried to go from this formula and use all the things I said in the Doubts. But I always get stuck when involves the functions in $C^{\infty}(M_1 \times M_2)$. Something seems not to fit. Anyone may help me ?