# Tagged Questions

**0**

votes

**1**answer

138 views

### Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R.
Is it possible to extend this function to a smooth function on R²?
...

**3**

votes

**1**answer

216 views

### Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...

**3**

votes

**1**answer

165 views

### Fundamental proof of the baby case of Hofer's theorem about displacement energy

In 1990, Hofer proved that the displacement energy of a standard ball in $C^{n}$ equals it's Gromov area.
Here is the baby case: Consider a smooth bounded function $f:R^{2}\rightarrow R$. Consider ...

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votes

**0**answers

343 views

### covariant derivative complex manifold

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...

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votes

**1**answer

207 views

### biholomorphism complex manifold induced structure

Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. ...

**26**

votes

**15**answers

7k views

### What is the Implicit Function Theorem good for?

What are some applications of the Implicit Function Theorem in calculus? The only applications I can think of are:
the result that the solution space of a non-degenerate system of equations ...

**93**

votes

**16**answers

13k views

### How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...