Dtjytyk

Let $M$ be a Riemannian manifold with Levi-Civita connection $ nabla$. Let $S$ be a differentiable manifold and $ varphi : S to M $ be a $C^{infty}$ immersion. Let
$$ D : TS times { text{vector fields along } varphi } to TM $$ s.t.

1. $ (v,X) mapsto D_v(X) in T_{varphi(pi(v))} M $




2. $D_{alpha v_1 + beta v_2}(X) = alpha D_{v_1}(X) + beta D_{v_2}(X)$



3. 
   $D_v( X+Y) = D_v(X)+D_v(Y)$ and $ D_v(fX) = f(pi(v))D_v(X) + v(f)X_{pi(v)}$
   


4. 
   $D_v (varphi^* (X) ) = nabla_{dvarphi_{pi(v)}(v)} (X) $ for any $X$ vector field in $M$.

where $ pi : TM to M$ is the natural projection $ v in T_pM mapsto p $ and $varphi^*(X) = X circ varphi $

I want to prove
$$ D_X( dvarphi(Y)) - D_Y( dvarphi(X)) = dvarphi ( [X,Y] )$$

Being $dvarphi(X)$ and $d varphi(Y)$ vector fields along $varphi$ I can write
$$ dvarphi(X) = U^i partial_i quad quad dvarphi(Y) = V^j partial_j $$
where $partial_i bigr |_p = frac{partial}{partial x^i} bigr |_{varphi(p)}$ for each $p in S$ and $ { x^1, dots x^n } $ are coordinates near $varphi(p)$ in $M$. By linearity and using Leibniz rule the LHS becomes
$$ V^jD_X(partial_j) + X(V^j) partial_j - U^iD_Y(partial_i) - Y(U^i) partial_i =$$
$$ = V^j U^i nabla_{frac{partial}{partial x^i}} (frac{partial}{partial x^j}) + X(V^j) partial_j - V^j U^i nabla_{frac{partial}{partial x^j}} (frac{partial}{partial x^i})- Y(U^i) partial_i$$
$$= X(V^j) partial_j - Y(U^i) partial_i$$

The RHS is, for any $p in S$ and $f in C^{infty}_{varphi(p)}M$
$$(dvarphi ( [X,Y] ))_p (f) = dvarphi_p ( [X,Y]_p) (f) = [X,Y]_p (f circ varphi) $$
$$= X_p(Y(f circ varphi)) - Y_p(X(f circ varphi)) = X_p(dvarphi(Y)(f)) - Y_p(dvarphi(X)(f))$$
$$= X_p( V^j frac{partial f}{partial x^j} bigr |_{varphi( ) } ) - Y_p( U^i frac{partial f}{partial x^i} bigr |_{varphi( ) } ) $$ $$= X_p(V^j)frac{partial f}{partial x^j} bigr |_{varphi(p) } - Y_p(U^i)frac{partial f}{partial x^i} bigr |_{varphi(p) } + V_p^jX_p ( frac{partial f}{partial x^j} bigr |_{varphi( ) }) - U_p^iY_p ( frac{partial f}{partial x^i} bigr |_{varphi( ) }) $$
$$= LHS + V_p^jX_p ( frac{partial f}{partial x^j} bigr |_{varphi( ) }) - U_p^iY_p ( frac{partial f}{partial x^i} bigr |_{varphi( ) }) $$

Where is my mistake?

There is no mistake. Actually the remaining term is zero:

We have

begin{align}V_p^jX_p ( frac{partial f}{partial x^j} bigr |_{varphi( ) })=V_p^jX_p ( frac{partial f }{partial x^j} circvarphi)
=V_p^j(dvarphi (X))_p ( frac{partial f}{partial x^j} )
=V_p^j U^i_pfrac{partial }{partial x^i}bigr |_{varphi(p) } ( frac{partial f}{partial x^j} )end{align}

Similar

$$U_p^iY_p ( frac{partial f}{partial x^i} bigr |_{varphi( ) }) =U_p^i V^j_pfrac{partial }{partial x^j}bigr |_{varphi(p) } ( frac{partial f }{partial x^i} )$$

so the remaining term is given by

$$V^j_pU^i_pleft(left[ frac{partial }{partial x^i}, frac{partial }{partial x^j}right]_{varphi(p)}(f)right)=0$$

since on canonical tangent vectors given by a coordinate chart the Lie bracket vanishes.