Intereting Posts

Finding the convergence radius of $\sum_{n=1}^{\infty}\frac{n!}{n^n}(x+2)^n$
Deeply confused about $\sqrt{a^5}=(a^5)^{1/5}$
What is an example that a function is differentiable but derivative is not Riemann integrable
Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?
Difference between Deformation Retraction and Retraction
Row reduction and the characteristic polynomial of a matrix
Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?
Prove that $A \subset B$ if and only if $A \cap B = A$
Areas versus volumes of revolution: why does the area require approximation by a cone?
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
Reducibility of polynomials over finite fields
Prove that the dihedral group $D_4$ can not be written as a direct product of two groups
Test for convergence with either comparison test or limit comparison test
Prove that there are infinitely many pairs such that $1+2+\cdots+k = (k+1)+(k+2)+\cdots+N$
Understanding branch cuts for functions with multiple branch points

I am not an expert in this so I hope this doesn’t sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for (homogenous) transformation group $\mathrm{SE}(3)$. For the latter I have seen use of Riemannian metric. For both I am guessing (as this is not my field) that there are some kind of manifold structure involved, so I can’t help to think that there are also good metrics (by good I mean metric that is considered “practical” when one considers rigid body motions).

Furthermore, is there a way to visualize the manifolds from $\mathrm{E}(3)$ and $\mathrm{SE}(3)$? Any good reference in this area?

- Can $S^4$ be the cotangent bundle of a manifold?
- The “Easiest” non-smoothable manifold
- Cohomology of projective plane
- Is there a characteristic property of quotient maps for smooth maps?
- Some questions about $S^n$
- Top homology of an oriented, compact, connected smooth manifold with boundary

- which axiom(s) are behind the Pythagorean Theorem
- Embed $S^{p} \times S^q$ in $S^d$?
- Why can't differentiability be generalized as nicely as continuity?
- Books on topology and geometry of Grassmannians
- Reference for self-intersections of immersions
- Is a connected sum of manifolds uniquely defined?
- Why the number of symmetry lines is equal to the number of sides/vertices of a regular polygon?
- Showing a diffeomorphism extends to the neighborhood of a submanifold
- Equivalence of two definitions of Spin structure
- Showing every knot has a regular projection using differential topology

There are several natural metrics that you can use on transformation groups. For example, for any topological group you could ask for a left or a right or a bi-invariant metric. Another natural way to define a metric is via how it acts on a space. If you really want to think about the isometry group of $\mathbb R^3$, I suppose this is the direction you’d want to go. So let $f$ and $g$ be isometries of $\mathbb R^3$. Define

$$d(f,g) = \max \{ |f(x)-g(x)| : x \in \mathbb R^3, |x| \leq 1 \}$$

This has the advantage that it’s left-invariant already. IMO this is a pretty good metric. If $f$ and $g$ fix the origin they’re linear and this is the norm of $f-g$, which is considered a fairly standard metric in linear algebra.

- Continuous images of Cauchy sequences are not necessarily Cauchy
- How to calculate Vapnik-Chervonenkis dimension
- $G$ is $p$-supersoluble iff $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.
- Product of repeated cosec.
- Flow of sum of non-commuting vector fields
- A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph
- Random point uniform on a sphere
- Does regular representation of a finite group contain all irreducible representations?
- How to solve this equation:$\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5$
- Conservation of mass in hyperbolic PDE
- Find the Remainder when $792379237923$…upto 400 digits is divided by $101$?
- Writing a function $f$ when $x$ and $f(x)$ are known
- What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
- Find an expression for the heat at the centre of the sphere, with temperature modelled by the given PDE
- Passing from induction to $\infty$