# Tagged Questions

**6**

votes

**1**answer

259 views

### Equivalence of exterior forms

Let us start with the following definition.
Let $1\leqslant k\leqslant n$ and let $\omega_1,\omega_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega_1$, $\omega_2$ are equivalent, if there exists ...

**1**

vote

**1**answer

168 views

### Distributing the Hodge map over the wedge product

Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to ...

**1**

vote

**0**answers

152 views

### Non-negative Quadratic forms with Exterior Forms

Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over ...

**3**

votes

**1**answer

181 views

### Decomposability of exterior two-forms

Hello,
The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in this context is ...

**8**

votes

**2**answers

829 views

### What is the canonical isomorphism between the tensor products of the top exterior powers associated to exact sequences of vector spaces?

One often reads (and writes) that an exact sequence of finite dimensional vector spaces
$$
0 \rightarrow X_1 \rightarrow X_2 \rightarrow \dots \rightarrow X_n \rightarrow 0
$$
induces a canonical ...

**6**

votes

**3**answers

1k views

### “Natural” pairings between exterior powers of a vector space and its dual

Let $V$ be a finite-dimensional vector space over a field $k$, $v_1, ... v_n \in V$ a set of vectors, and $f_1, ... f_n \in V^{\ast}$ a set of covectors. Up to permutation, there seem to be at least ...

**20**

votes

**6**answers

3k views

### Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find ...

**16**

votes

**12**answers

5k views

### Geometrical meaning of Grassmann Algebra

I don't understand wedge product and Grassmann algebra. However, I heard that these concepts are obvious when you understand the geometrical intuition behind them. Can you give this geometrical ...