# How Can Wedge Products Be Used in Differential Geometry?

• I
• Silviu
In summary: The reason for this is that wedge products of one-forms are anti-symmetric, which is necessary for a 2-form.
Silviu
Hello! I am reading something about differential geometry and I have that for a manifold M and a point ##p \in M## we denote ##\Omega_p^r(M)## the vector space of r-forms at p. Then they say that any ##\omega \in \Omega_p^r(M)## can be expanded in terms of wedge products of one-forms at p i.e. ##\omega =\frac{1}{r!}\omega_{\mu_1 ...\mu_r}dx^\mu_1 \wedge dx^\mu_2 ... \wedge dx^{\mu_r}##, with ##\omega_{\mu_1 ...\mu_r}## completely antisymmetric. I am not sure why. If I have a 2-form, that would be ##\omega=\omega_{\mu\nu}dx^\mu dx^\nu##, but the ##\omega_{\mu\nu}## and ##\omega_{\nu\mu}## don't need to me in any relationship (equal or opposite), so how can I use the wedge product which would basically contain ##dx^\mu dx^\nu-dx^\nu dx^\mu## to obtain it? Thank you!

An r-form is defined as a completely anti-symmetric type (0,r) tensor so indeed ##\omega_{\mu_1 \ldots \mu_r}## will be the components of that anti-symmetric tensor.

Now, since the wedge product basis is anti-symmetric, adding something symmetric to the components will actually not matter at all (its contribution to the sum would be zero), but seen as just a type (0,r) tensor, its component would be anti-symmetric.

Note that ##\omega = \omega_{\mu\nu} dx^\mu \otimes dx^\nu## is not a 2-form, since it is not anti-symmetric and r-forms are anti-symmetric by definition.

Silviu said:
...If I have a 2-form, that would be ##\omega=\omega_{\mu\nu}dx^\mu dx^\nu##...

It would be ##\omega=\omega_{\mu\nu}dx^\mu \wedge dx^\nu##

## 1. What is the Wedge Product in mathematics?

The Wedge Product is a mathematical operation used in differential geometry and algebraic topology. It is a way to combine two vectors or forms to create a new vector or form that captures the geometric essence of their relationship.

## 2. How is the Wedge Product denoted?

The Wedge Product is denoted by the symbol ∧ (a caret with a tilde on top). For example, the Wedge Product of two vectors x and y would be written as x ∧ y.

## 3. What is the difference between r-forms and k-forms?

R-forms and k-forms are both types of differential forms, which are mathematical objects that generalize the concept of a vector. The main difference between them is that r-forms are defined on r-dimensional manifolds, while k-forms are defined on k-dimensional manifolds. In other words, r-forms are defined on higher-dimensional spaces than k-forms.

## 4. How is the Wedge Product used in differential geometry?

In differential geometry, the Wedge Product is used to define the exterior derivative, which is a way to differentiate differential forms. It is also used to define the Hodge star operator, which is used to convert between different types of forms. Additionally, the Wedge Product is used to define the wedge product of tensors, which is used in the study of curvature and other geometric properties of manifolds.

## 5. Can the Wedge Product be extended to more than two vectors or forms?

Yes, the Wedge Product can be extended to any number of vectors or forms. For example, the Wedge Product of three vectors x, y, and z would be written as x ∧ y ∧ z. This extension is useful in many mathematical applications, including multilinear algebra and the study of differential forms on higher-dimensional manifolds.

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