Intereting Posts

How to prove that no constant can bound the function f(x) = x
Indian claims finding new cube root formula
Solving an infinite series containing $\arctan$
What is the minimal number of generators of the ideal $(6x, 10x^2, 15x^3)$ in $\Bbb Z$?
Distribution of $-\log X$ if $X$ is uniform.
Show that $3p^2=q^2$ implies $3|p$ and $3|q$
Topologically distinguishing Mobius Strips based on the number of half-twists
Mathematically rigorous text on classical electrodynamics.
Is there something like Cardano's method for a SOLVABLE quintic.
Asymptotic of a sum involving binomial coefficients
Proving that the coefficients of the characteristic polynomial are the traces of the exterior powers
Differentiating $4^{x^{x^x}}$
Geometry of the dual numbers
Eigenvalues of a matrix with binomial entries
What is the difference between writing f and f(x)?

This problem was given to me by a friend:

Prove that $\Pi_{i=1}^m \mathbb{S}^{n_i}$ can be smoothly embedded in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$.

The solution is apparently fairly simple, but I am having trouble getting a start on this problem. Any help?

- Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds
- How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?
- Homotopically trivial $2$-sphere on $3$-manifold
- metric on the Euclidean Group
- Proof the degree of a reflection through a hyperplane is −1.
- Why any short exact sequence of vector spaces may be seen as a direct sum?

- Are closed orbits of Lie group action embedded?
- Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.
- Obstructions to lifting a map for the Hopf fibration
- Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory
- When is a fibration a fiber bundle?
- Is every embedded submanifold globally a level set?
- Can $S^4$ be the cotangent bundle of a manifold?
- Equivalent condition for non-orientability of a manifold
- A manifold such that its boundary is a deformation retract of the manifold itself.
- What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

**EDIT:** I **cannot** delete this post as it’s been already accepted 🙁

The proof is carried out by induction on $m.$

$m=1$ is trivial by choosing coordinates $(x^{(1)}_0,x^{(1)}_1,…,x^{(1)}_{n_1})$ where $\sum_{j=0}^{n_1} (x_j^{(1)})^2=1$, so let $m=2,$ then similarly $S^{n_1} \times S^{n_2}$ is embedded in $\mathbb{R}^{n_1+n_2+2}$ which also lies in the hypersurface $H$ with equation $\sum_{j=0}^{n_1} (x_j^{(1)})^2+\sum_{j=0}^{n_2} (x_j^{(2)})^2=2.$ In fact, $H$ is diffeomorphic to $S^{n_1+n_2+1}.$ Now, the embedding is missing at least a point, for example $p=(0,…,0,1,1) \in H,$ so by stereographic projection $S^{n_1+n_2+1} \setminus \{p\}$ is *diffeomorphic* to $\mathbb{R}^{n_1+n_2+1}.$

Suppose that the assertion holds for $<m,$ then $(S^{n_1}\times…\times S^{n_{m-1}}) \times S^{n_m}$ is embedded diffeomorphically in $\mathbb{R}^{n_1+…+n_{m-1}+1} \times \mathbb{R}^{n_m+1}\cong \mathbb{R}^{n_1+…+n_{m}+2}$ hence following the same argument you can reduce the dimension by $1,$ so the result.

- Note first that $\mathbb{R}\times\mathbb{S}^n$ smoothly embeds in $\mathbb{R}^{n+1}$ for each $n$, via $(t,\textbf{p})\mapsto e^t\textbf{p}$.
- Taking the Cartesian product with $\mathbb{R}^{m-1}$, we find that $\mathbb{R}^m\times\mathbb{S}^n$ smoothly embeds in $\mathbb{R}^{m}\times\mathbb{R}^n$ for each $m$ and $n$.
- By induction, it follows that $\mathbb{R}\times\prod_{i=1}^m \mathbb{S}^{n_i}$ smoothly embeds in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$.

The desired statement follows.

- A question about two theories and their models
- Prove that 10101…10101 is NOT a prime.
- A good Open Source book on Analytic Geometry?
- Integrating $\int_0^ex^{1/x}\ \mathrm dx$
- Taking limits on each term in inequality invalid?
- Legendre symbol: Showing that $\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0$
- Absolutely continuous maps measurable sets to measurable sets
- Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$
- How can a structure have infinite length and infinite surface area, but have finite volume?
- Cardinality of Irrational Numbers
- Implicit function theorem for several complex variables.
- A polynomial sequence
- When exactly is the splitting of a prime given by the factorization of a polynomial?
- Countable product of Polish spaces
- Epimorphisms from a free group onto a free group