MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
    
Your title mentions almost-complex manifolds, but your question doesn't. – L Spice Mar 27 at 13:50
up vote 7 down vote accepted

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 $$\left<Z_1,Z_2\right>=\left<X_1,X_2\right>+\left<Y_1,Y_2\right>+i\left<Y_1,X_2\right>-i\left<X_1,Y_2\right>$$.

share|cite|improve this answer
1  
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 at 14:49
1  
It is the real part of the Hermitian one, so it is a Riemannian metric, not complex-valued. – Ben McKay Mar 27 at 14:52
    
@JoséFigueroa-O'Farrill I need the common definition such that in books is used. But I could not find a definition that describes such a metric and such a derivative. – Baghban Mar 27 at 15:25
    
@BenMcKay Are these definitions the common ones? – Baghban Mar 27 at 18:22
    
The are the common definitions, except that one usually uses Hermitian metrics on complex vector bundles. – Ben McKay Mar 27 at 19:02

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.