# Tag Info

Accepted

### Necessary and sufficient mathematical structure for spacetime continuum

First, let me remark that there is no such thing as a “necessary and sufficient condition” for modelling spacetime. It’s a model, so it can’t be an exact description (and Physics doesn’t answer the ...
• 54.2k
Accepted

### Is smoothness of multiplication redundant in the definition of Lie Group?

As suggested, I’m turning my comment (I was worried I was missing something) into an answer. The statement is wrong. Consider the classical manifold $\mathbb{R}$, with the non-smooth “addition” law ...
• 34.2k

• 10.9k

### Relation between trilinear forms, Jacobi identity and closed $2$-forms

The relevant formula is the "conceptual" formulation of the exterior derivative. If $\omega$ is a $2$-form and $X,Y,Z$ are vector fields (here, on $\Bbb R^3$, we have \begin{multline*} d\...
• 114k
Accepted

### Why is the signature of a manifold homotopy invariant?

The definition you've given for the signature is phrased in terms of de Rham cohomology and is indeed not obviously homotopy invariant. However, the pairing you've given has a "lift" to ...
• 11.6k

### Reference: the number of smooth structures on a topological $n$-manifold, $n\geq 5$ is finite

There are not really as many structures on $T^n$ as that: typically one mods out $H^3(T^n;\mathbb{Z}/2)$ by the action of $GL(n,\mathbb{Z}/2)$. This is discussed by Wall in chapter 15A of Surgery on ...

### Can every Lie group be realized as conformal group of smooth manifolds

First of all, one cannot talk about conformal transformations of a smooth manifold: For the notion of conformality to be defined you need an extra structure besides a smooth atlas. I will work with ...
• 95.7k
Accepted

### Show a operator is not a tensor field on $\mathbb{R}^3$

Note: I found this one quite hard to explain, and ended up making two major edits. Edit 1 was to highlight that the crux of the matter is that $T|_x(v_1, v_2, v_3)$ depends on the partial derivatives ...
• 32.1k

### The embedded submanifolds of a smooth manifold (without boundary) of codimension 0 are exactly the open submanifolds

There is a proposition that characterizes embedded submanifolds of codimension $k$ as follows. Let $N\subset M$ denote a codimension $k$ embedded submanifold of $M$. If $p\in N$, then there exists a ...
• 26.1k
### Calculating the differential of the map $A \mapsto A A^T$, $A \in M(n\times n, \mathbb{R})$
Regarding your 2nd question: the tangent space of the space of matrices $M(n\times n, \mathbb R)$ at a point $A$ is, as you said, identified with $M(n\times n, \mathbb R)$ itself. Regarding your 1st ...