Intereting Posts

The Helmholtz equation: How prove this $T\psi{(x)}\in\Omega$.
Radius of convergence of Taylor series of holomorphic function
What does the math notation $\sum$ mean?
Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational
$I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible
Double Complement of a set proof
Rationalizing the denominator of $\frac {\sqrt{10}}{\sqrt{5} -2}$
Rank of sum of rank-1 matrices
Metric spaces problem
Requesting abstract algebra book recommendations
Property of short exact sequences.
Does $L^p$-convergence imply pointwise convergence for $C_0^\infty$ functions?
What are some applications outside of mathematics for algebraic geometry?
To characterize uncountable sets on which there exists a metric which makes the space connected
Diagonalizable properties of triangular matrix

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension.

So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples?

- Fix point of $L:S^2\rightarrow S^2$
- Why is the geometric locus of points equidistant to two other points in a two-dimensional Riemannian manifold a geodesic?
- Definition of vector field along a curve
- Computation in Wikipedia's article “Riemann Curvature Tensor”
- How to Classify $2$-Plane Bundles over $S^2$?
- No Smooth Onto Map from Circle to Torus
- How to show that a diagram is a pushout in the category $\text{TOP}$?
- weak homotopy equivalence (Whitehead theorem) and the *pseudocircle*
- shortest distance between two points on $S^2$
- Pushforward of Inverse Map around the identity?

This question was essentially answered here, namely, that the cut-locus has measure zero (see the references provided in the link). I assume that your manifold $M$ is smooth, otherwise, I am not sure what notion of measure zero you would be using. I will also assume that $M$ is connected. (If not, apply this argument to each connected component.) Then, put a complete Riemannian metric on $M$, consider the cut-locus $C(p)$ of a point $p\in M$ and the open subset $U(p)=M\setminus C(p)$. The exponential map $\exp_p: T_pM\to M$ restricts to a diffeomorphism $V(p)\to U(p)$, where $V(p)\subset T_pM$ is a certain open subset (diffeomorphic to the open $n$-ball, where $n$ is the dimension of $M$).

One can ask a similar question in the context of connected $n$-dimensional topological manifolds $M$. Instead of removing a measure zero set, one can remove a closed nowhere dense subset. Then in all dimensions but 4 it follows from the results in the book by Kirby and Siebenmann that $M$ contains an open subset $U$ homeomorphic to $R^n$. (Maybe it was also known prior to their work, I am not sure.) I do not know what to say about dimension 4.

One last thing, Sullivan proved that every topological manifold $M$ of dimension $\ne 4$ admits a unique *Lipschitz structure*, i.e., an atlas where transition maps are locally Lipshitz. This implies that topological $n$-manifolds ($n\ne 4$) have a well-defined notion of measure zero sets (very indirect though). However, I do not know if the residual set in Kirby-Sibenmann will have measure zero in this sense, this goes well beyond my understanding of the work of their work and the one by Sullivan.

**Edit.** A better reference for topological manifolds (instead of Kirby-Siebenmann) is in my answer here, to papers by Brown (compact case) and Berlanga (noncompact case).

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