On the complexification of a Riemannian manifold

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.

• Your title mentions almost-complex manifolds, but your question doesn't. – LSpice Mar 27 '16 at 13:50

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 $$\left<Z_1,Z_2\right>=\left<X_1,X_2\right>+\left<Y_1,Y_2\right>+i\left<Y_1,X_2\right>-i\left<X_1,Y_2\right>$$.
• There's something odd about your metric: it's neither the hermitian nor the bilinear extension of $g$. Of course, it's not clear what the OP wants. Usually when people talk about complexifying the riemannian manifold, it means extending the metric complex-bilinearly. This is then usually followed by a restricting it on some other real section of the complexified tangent bundle. – José Figueroa-O'Farrill Mar 27 '16 at 14:49