Intereting Posts

Perpendicular Bisector of Made from Two Points
Does an injective endomorphism of a finitely-generated free R-module have nonzero determinant?
Group of order 24 with no element of order 6 is isomorphic to $S_4$
How do I add summation formula's like this: $F(n+2) = 1 + \sum\limits_{i=0}^{n} F(i)$?
Modules with projective dimension $n$ have not vanishing $\mathrm{Ext}^n$
Creating surjective holomorphic map from unit disc to $\mathbb{C}$?
Fitting a sine function to data
How is $\mathbb{C}$ different than $\mathbb{R}^2$?
Curve of a fixed point of a conic compelled to pass through 2 points
Drawing subgroup diagram of Dihedral group $D4$
Prove that every group of order $4$ is abelian
Evaluate $\prod_{n=1}^\infty \frac{2^n-1}{2^n}$
Any even elliptic function can be written in terms of the Weierstrass $\wp$ function
The set of convergence of a sequence of measurable functions is measurable
Is $i = \sqrt{e^{\pi\sqrt{e^{\pi\sqrt\ldots}}}}$?

Is there a criterion to show that a level set of some map is not an (embedded) submanifold? In particular, an exercise in Lee’s smooth manifolds book asks to show that the sets defined by $x^3 – y^2 = 0$ and $x^2 – y^2 = 0$ are not embedded submanifolds.

In general, is it possible that a level set of a map which does not has constant rank on the set still defines a embedded submanifold?

- (Locally) sym., homogenous spaces and space forms
- The integral of a closed form along a closed curve is proportional to its winding number
- Do there exist manifolds which cannot be smoothly embedded in a compact manifold?
- precise official definition of a cell complex and CW-complex
- Is the connected sum of complex manifolds also complex?
- Applications of Whitney's Approximation Theorem

- What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration?
- Folliation and non-vanishing vector field.
- Homeomorphic or Homotopic
- Is there an easy way to show which spheres can be Lie groups?
- For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?
- concrete examples of tangent bundles of smooth manifolds for standard spaces
- Embedding a manifold in the disk
- Hairy Ball Theorem: homotopy from the identity to the antipodal map
- precise official definition of a cell complex and CW-complex
- Does having a codimension-1 embedding of a closed manifold $M^n \subset \mathbb{R}^{n+1}$ require $M$ to be orientable?

It is certainly possible for a level set of a map which does not have constant rank on the set to still be an embedded submanifold. For example, the set defined by $x^3 – y^3 = 0$ is an embedded curve (it is the same as the line $y=x$), despite the fact that $F(x,y) = x^3 – y^3$ has a critical point at $(0,0)$.

The set defined by $x^2 – y^2 = 0$ is not an embedded submanifold, because it is the union of the lines $y=x$ and $y=-x$, and is therefore not locally Euclidean at the origin. To prove that no neighborhood of the origin is homeomorphic to an open interval, observe that any open interval splits into exactly two connected components when a point is removed, but any neighborhood of the origin in the set $x^2 – y^2$ has at least four components after the point $(0,0)$ is removed.

The set $x^3-y^2 = 0$ is an embedded topological submanifold, but it is not a smooth submanifold, since the embedding is not an immersion. There are many ways to prove that this set is not a smooth embedded submanifold, but one possibility is to observe that any smooth embedded curve in $\mathbb{R}^2$ must locally be of the form $y = f(x)$ or $x = f(y)$, where $f$ is some differentiable function. (This follows from the local characterization of smooth embedded submanifolds as level sets of submersions, together with the Implicit Function Theorem.) The given curve does not have this form, so it cannot be a smooth embedded submanifold.

The set given by $(x^2+y^2)(x^2+y^2-1)=0$ is an embedded submanifold in $\mathbb R^2$

but it has components of different dimension and so I guess the map does not have constant rank on the set, but I haven’t checked.

- How can I prove that an order (“$<$” say) on $\mathbb Z_n$ cannot be defined?
- Rudin's 'Principle of Mathematical Analysis' Exercise 3.14
- Generalised Binomial Theorem Intuition
- totally disconnected orbit-stabilizer theorem
- Is this statement always true?
- Infinite metric space has open set $U$ which is infinite and its complement is infinite
- If $f$ is a function of slow increase, then $f$ is slowly varying?
- Find the limit of $x_n^3/n^2$ if $x_{n+1}=x_{n}+1/\sqrt{x_n}$
- Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$
- What are the convergent sequences in the cofinite topology
- Positive lower bound for a function defined as a series?
- Positive Linear Functionals on Von Neumann Algebras
- Absolute Value inequality help: $|x+1| \geq 3$
- Exact sequence arising from symplectic manifold
- Find and classify the bifurcations that occur as $\mu$ varies for the system