Articles of smooth manifolds

Is a map with invertible differential that maps boundary to boundary a local diffeomorphism?

Let $M,N$ be smooth manifolds with boundary (of the same dimension). Let $f:M \to N$ be a smooth map satisfying $(1) \, \,f(\partial M)=\partial N,f(\operatorname{Int} M)=\operatorname{Int} N$. $(2) \, \, df_p$ is invertible for every $p \in M$. Is it true that $f$ is a local homeomorphism? I suspect $f$ must in fact be a […]

Free and proper action

I don’t know how to solve this problem. Let G be a Lie group and H a closed Lie subgroup ,that is, a subgroup of G which is also a closed submanifold of G. Show that the action of H in G defined by A(h,g)=h.g is free and proper. Can you please define what is […]

Special linear group as a submanifold of $M(n, \mathbb R)$

I have that $SL(n,\mathbb{R})$ is an embedded submanifold of dimension $n^2-1$ in $GL(n,\mathbb{R})$, and I know that $T_XGL(n,\mathbb{R})$ is isomorphic to $M(n,\mathbb{R})$ for all $X \in GL(n , \mathbb R)$. Is there a way I can use this to show that $SL(n,\mathbb{R})$ is a smooth submanifold of $M(n,\mathbb{R})$ and get is dimension? Otherwise, how could […]

What Does This Notation Mean (“derivative” of a 1-form)?

On page 47 of Helgason’s book Differential Geometry, Lie Groups, and Symmetric Spaces, he uses the notation $$\frac{\partial \omega}{\partial t}$$ where $\omega$ is a 1-form on a manifold and $t$ is one of the coordinates of a chart. (If you look on the page, you will see that $\omega$ actually has super and subscript indices […]

Show that $\mathbb{B}^n $ is a smooth manifold with its boundary diffeomorphic to $S^{n-1}$

Consider a closed unit ball $\mathbb{B}^n = \{ x\in\mathbb{R^n} : \|x\|\le 1\} $ How do I show that $\mathbb{B}^n $ is a smooth manifold with its boundary ($\partial \mathbb{B^n}$) diffeomorphic to $S^{n-1}$ ?

How To Formalize the Fact that $(g, h)\mapsto dL_g|_h$ is smooth where $g, h\in G$ a Lie Group

Let $G$ be a Lie group. I am wondering if there is a way to say that the map $(g, h)\mapsto dL_g|_h$ defined on $G\times G$ is a smooth map (Here $L_g$ is the left translation map from $G$ to $G$ and by $dL_g|_h$ I mean the differential of $L_g$ at $h$). The challenge here […]

Is the following set a manifold?

Show (using the implicit function theorem) that the following subset $$M:=\{(x,y,z)\in\mathbb{R}^{3}\;|x^2+y^4+z^4=3\}\subseteq\mathbb{R}^{3}$$ Theorem: Let $A\subset \mathbb{R}^{n}$ be open let $g:A\to \mathbb{R}^{p}$ be a differentiable function such that $g^{\prime}(x)$ has rank $p$ whenever $g(x)=0$. Then $g^{-1}(0)$ is an $n-p$ dimensional manifold in $\mathbb{R}^{n}$ Solution. Let $F(x,y,z)=x^2+y^4+z^4-3$, denote $M=F^{-1}(0)$. Then for each $P=(x,y,z)\in M$ we have $$F^{\prime}(x,y,z)=\nabla F(x,y,z)=(2x,4y^3,4z^3)\neq […]

When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I want to know under what conditions the universal cover $\tilde{M}$ is complete. The reason for this questions is that I want to know under what conditions […]

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for $t \in [0, \varepsilon]$, $h(t) = 1 − \frac{1}{\ln t}$ for all $t$ large enough. (You don’t have […]

The identity map on a tangent space

If $X$ is a smooth manifold and $I : X \rightarrow X$ is the identity map on $X$ (with the same smooth structure on both sides) then I can show that $dI_x : T_x X \rightarrow T_x X$ is the identity map on the tangent space $T_x X$ of $X$ at some point $x \in […]