Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $TM \otimes\mathbb{C}$ as what there exists and how can we do the differentiation from the sections of this bundle by using the Levi-Civita connection of the metric $g$?


I mean If $X, Y$ are two sections of $TM \otimes\mathbb{C}$ then the inner product of $X, Y$ how can be defined by using the metric $g$?


Moreover, how can we differentiate from $X$ along $Y$ in a natural way by using the Levi-Civita connection of $g$?


I can not find a definition which describes such a metric and differentiation.



On the complexification of a Riemannian manifold
Each section of $TM \otimes \mathbb{C}$ has a unique decomposition $Z=X+iY$ as a sum with $X$ and $Y$ sections of $TM$. Define your metric using this, for example as $\left<Z\_1,Z\_2\right>=\left<X\_1,X\_2\right>+\left<Y\_1,Y\_2\right>$. Use an affine connection as $\nabla\_{X+iY} U+iV=\nabla\_X U - \nabla\_Y V + i \left(\nabla\_Y U + \nabla\_X V\right)$ to get complex linearity.


Edit: the natural Hermitian metric on $TM \otimes \mathbb{C}$ is $$\left<Z\_1,Z\_2\right>=\left<X\_1,X\_2\right>+\left<Y\_1,Y\_2\right>+i\left<Y\_1,X\_2\right>-i\left<X\_1,Y\_2\right>$$.


