# Differentisl geometry internal contraction

In summary, the conversation discusses how the relation (df:U)(dh:V)-(dh:V)(df:V) (1) can be rewritten as U(fdh:v)-V(fdh:U)-fdh:[U,V] (2). The speaker is struggling to understand the process of rewriting and has tried expanding (1) by making a scalar vector product, but is unsure if this is correct. They ask for clarification on the variables f, h, U, and V, as well as the use of the : symbol.
Hi
I am struggling to see how the relation:
$(df:U)(dh:V)-(dh:V)(df:V)$ (1)
can be rewritten as:
$U(fdh:v)-V(fdh:U)-fdh:[U,V]$ (2)

I have tried expanding (1) out by making a scalar vector product of Udh:V but i don't think that is right and i am just not sure how best to proceed. It was annoyingly skipped over in class today, any help or suggestions would be appreciated.

I am very unclear as to what you are asking. Perhaps you could clarify? What is f,h,U,V? What does your : mean? Typically, I have seen contraction denoted by $\iota$ or $\lrcorner$, so I do not know how to read this.

## What is differential geometry internal contraction?

Differential geometry internal contraction is a mathematical concept that involves the contraction of a differential form or tensor field on a differentiable manifold. It is used to simplify and study the geometry of a manifold by reducing the number of independent variables.

## How is differential geometry internal contraction used in physics?

Differential geometry internal contraction is used in physics to describe the behavior of physical systems in terms of geometric properties. It is particularly useful in general relativity, where it is used to express the curvature of spacetime in terms of a metric tensor field.

## What are the applications of differential geometry internal contraction?

Differential geometry internal contraction has applications in various fields such as differential equations, differential geometry, physics, and engineering. It is used to study the geometry of spaces with non-Euclidean properties, such as curved surfaces and spacetime.

## What is the difference between internal and external contraction in differential geometry?

The main difference between internal and external contraction in differential geometry is the direction of the contraction. Internal contraction involves the contraction of indices within a tensor, while external contraction involves the contraction of indices between two tensors.

## Are there any practical examples of differential geometry internal contraction?

Yes, there are many practical examples of differential geometry internal contraction, such as the use of the Ricci tensor in general relativity to describe the curvature of spacetime, or the use of the Laplace operator in the heat equation to describe the change in temperature over time.

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