# Tagged Questions

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### Manifold locally looks like a open set but not as a euclidean space?

I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define ...
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### Unique manifold structure

I am reading the first chapter from the book - Foundations of Differentiable manifolds and Lie groups by Warner. There, he has given two statements to be proved as exercises. a) Let $M$ be a ...
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### Partition of Unity Argument - Please help me bridge the two sides of what I've done so far.

Let $M$ be a compact n-dimensional manifold. Show the following: $\displaystyle \int {e^{w} \sqrt{g}} dx \leq C \exp(c ||Dw||_{L^{n}}^{n} + ||w||_{L^{n}}^{n})$, where $g$ is the determinant of ...
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### Why does the expression of a point on a manifold seem to assume a coordinate system?

This question came up when I was studying the definition of natural coordinate functions. In many books, such as O'Neil, natural coordinate functions are defined as $u_i(p) = p_i$ where ...
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### Nonvanishing vector fields on $\mathbf{S}^2$ and bases for $T_p\mathbf{S}^2$.

A vector field on a manifold is a (continuous, differentiable) map $X: M \to TM$ such that $X(p) \in T_pM$ for each $p \in M$. The tangent space $T_pM$ has a basis. Take $M = \mathbf{S}^2$. For each ...
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### Showing that a map $h:S^2\rightarrow \mathbb{R}^4$ is an immersion

The Problem Let $h:S^2\rightarrow \mathbb{R}^4$ be a smooth map of the form $$h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$ Show that $h$ is an immersion for any $a,b\in \mathbb{R},a,b\neq 0,ab<0$. Attempt ...
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### Unitary group and unit circle

Let $U(n)$ denote the group of complex unitary matrices, let $S^1$ be the unit circle in the complex plane. Then the map $$f:U(n)\to S^1,\quad f(A)=det(A)$$ is a group homomorphism and a submersion. ...
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### Related to tangent space of points on the sphere

Let $S^n$ be the $n-$dimensional unit sphere. Define $h: S^3\to S^2$ and $F:S^1\times S^3\to S^3$ as follows $$h(x)=(2(x_1x_3+x_2x_4),2(x_2x_3-x_1x_4),(x_1^2+x_2^2)-(x_3^2+x_4^2))$$ ...
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### Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic ...
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### How to motivate vectors as derivations?

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space ...
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### Boundary orientation of $H^k$

$H^k$ is oriented by the standard orientation of $R^k$. Thus $\partial H^k$ acquires a boundary orientation. But $\partial H^k$ may be identified with $R^{k-1}$. Show that the boundary orientation ...
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### Intersection of two open sets in the projective plane

I want to compute the cohomology groups of the real projective plane, $P^2$, using Mayer Vietoris exact sequence. Now $H^0(P^2)=\mathbb{R}$, $H^2(P^2)=0$ being $P$ not orientable, so my problem really ...
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### Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
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### vector field on $\mathbb{R} P^2$

Actually this is a quesion in Lee's book, Manifolds and differential geometry. I have problems working with projective spaces as manifolds.(e.g. what are curves in projective spaces ? I need to know ...
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### Orientation of $X \times Y$

Suppose that $X$ is not orientable. How can I show that $X \times Y$ is never orientable, no matter what manifold $Y$ may be? I've tried supposing that $X \times Y$ is orientable, then using that ...
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### Degree of polynomial seen as a smooth map

I need some help with a part of an exercise. Let $P$ be a real polynomial of degree $d$, seen as a map $P:\mathbb{R}\rightarrow\mathbb{R}$. Prove that if $d$ is even then the degree of $P$, $degP$, ...
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### If a smooth map between manifolds is injective, is the induced map on the tangent spaces injective too?

If $\phi:M\longrightarrow N$ is an injective smooth map between two manifolds, then is $d\phi_m:M_m\longrightarrow N_{\phi(m)}$, the induced map between the tangent spaces injective too? I tried the ...
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### Which integral curves of a field are defined for all times t?

Which integral curves of the field $X=x^2 \frac{\partial}{\partial x} + y \frac{\partial}{\partial y}$ are defined for all times t? I would be very thankful if somebody can help me understand what ...
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### Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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### Relative Compactness $\Rightarrow$ Compactness

I try to figure out: $(\overline{A}^U\text{ compact in }U )\Rightarrow( \overline{A}^X\text{ compact in }X)$ ...while $U\in\mathcal{T}$ It's clear for the case: $\overline{A}^X\subseteq U$ But else, ...
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### Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
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### Curve in $\mathbb{P}^{n}(\mathbb{R})$, differentiable manifolds

I need a book which contains the demonstration that if $\Gamma$ is a curve in $\mathbb{P}^{n}(\mathbb{R})$ defined as $F(x_{1}, \cdots \ \ , x_{n+1}) = 0$ , with $F$ homogeneous polynomial, then ...
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### $C^k$ hypersurfaces can be split in this way?

Let $S$ be a bounded $C^k$ hypersurface of dimension $n \geq 2$ in $\mathbb{R}^{n+1}$ with no boundary. Is it true that $S$ can be split into two hypersurfaces $S_1, S_2$ that have boundary, and a ...
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### What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
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### Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
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### Dependence on the class of differentiability between manifolds and maps

Maybe a silly question, but in some books (like "Differential Geometry - Manifolds, Curves, Surfaces - Gostiaux and Berger"), when differentiable maps of class $C^s$ are defined, we have something ...
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### Integral curves of $X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}$ on a cylinder

Consider coordinates $(\theta, z)$ on $S^1 \times \mathbb R$, and a vector field $$X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}.$$ Show that the integral curve of ...
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### Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some ...