# Tagged Questions

For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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### Equivalent definition of properly discontinuous action

In the book An Introduction to Differentiable Manifolds and Riemannian Geometry by Boothby in Chapter $3$ the author gives the following definition: Definition($8.1$) A discrete group $\Gamma$ ...
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### Uniqueness of dimension of regular submanifold

Suppose $N$ is a manifold of dimension $n$. Now a regular submanifold $S$ of $N$ of dimension $k$ is defined as, if for every point $p$ of $S$ there is a coordinate chart $(U,u_*)$ from a maximal ...
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### Proof for showing that a set of space curves form a manifold

I basically have a smooth space curve $\alpha$,with curvature $\kappa$ and $\tau$, both non-zero, and I generate a family of curves $M_{\alpha} = \{\dfrac{\alpha}{\mu} : \mu \in (0, \infty) \}$ . The ...
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### Trajectories of vector fields on compact manifolds

Suppose that $X$ is a smooth vector field on a smooth manifold $M$. The trajectories of $X$ are curves $p(t)$ in $M$ which satisfy $d{p(t)}/{dt} = X(p(t))$. It's well known that $p(t)$ exists ...
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### Is the set $x^2-y^2 = 0, z > y \geq0$ a smooth manifold in $\mathbb{R}^3$?

Is the set $x^2-y^2 = 0, z > y \geq0$ a smooth manifold in $\mathbb{R}^3$? I think that the answer is no, but I'm not really sure how to prove this as I'm having trouble visualizing how it looks. ...
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### Necessary condition for the intersection of two submanifolds to be a submanifold

Let $X$ be an $n$ dimensional manifod. How could I prove that for arbitrary submanifolds $M,N$ of dimension $n-m,n-k$, if $\forall x\in M\cap N$ $dim( T_xM\cap T_xN)=n-m-k$ then $M\cap N$ is a ...
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### Evaluate $\int_Mxdy\wedge dz$ where $M$ is the torus formed by the circle of radius $1$ in the $xz$ plane centered at $(2,0,)$ rotated around $y$ axis

Evaluate $\int_Mxdy\wedge dz$ where $M$ is the torus obtained by rotating the circle $(x-2)^2+z^2=1$ around the $y$ axis. I've parameterized $M$ using $\alpha:(0,2\pi)\times (0,2\pi)\rightarrow M$ ...
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### Using Fubini's and Fundamental Theorem, explain why $\int_Sdx\wedge dy$=$\int_{\partial S}xdy$ where $S=\{(x,y,z)\in\mathbb R^3|x^2+y^2+z^2=1,z>0\}$

This is sort of an odd question because it's asking to justify generalized Stokes' Theorem using only "Fubini's" (i.e. equality of iterated integrals) and Fundamental Theorem of Calculus (i.e. 1-...
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### Describe an atlas of smoothly related charts for the Special Orthogonal Group $SO(3)$

The Special Orthogonal Group $SO(3) =$ {${A \in M_3(\mathbb{R}) : A^TA = I, det(A) = 1}$} I have successfully shown that $SO(3)$ is a manifold, but I am having a difficult time explicitly finding a ...
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### Parametrize $\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1, x\geq0\}$

Let $M =\{(x,y,z)\in\mathbb{R}^3\mid x+y^2+z^2=1,x\geq0\}$. It seems to me that this manifold is a "cone" since we have $y^2+z^2=1-x$ for $x\in[0,1]$ which, geometrically, is a circle in the $yz$ ...
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### Find the volume of h(P) in terms of volume of P

Where h:$\mathbb R^n \rightarrow R^n$ is the function h(x) = $\lambda x$ and P is a k-dimensional parallelopiped in $\mathbb R^n$ Attempt at the solution : Let $x_1,....,x_k$ be vectors in $R^n$ ...
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### Question on the definition of outward normal vector from Spivak, Calculus on Manifolds

The following definition of the outward unit normal at the boundary of a manifold $M \subseteq \mathbb R^n$ is taken from Spivak, Calculus on manifolds (page 119). If $M$ is a $k$-dimensional ...
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### The defintion of orientation of a manifold from Spivak, Calculus on Manifolds

In Spivak Calculus on Manifolds the author uses a definition of orientation of a manifold which I do not understand, and which I do not found elsewhere. I cite: It is often necessary to choose an ...
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### Counterexample of the fundamental theorem for hypersurfaces in the Euclidean space $\mathbb{R}^{n+1}$ in global sense

Are there any example of a closed Riemannian $n$-manifold $(M,g)$ and a symmetric bilinear form $A=h_{ij}dx^i\otimes dx^j\in\Gamma(T^\ast M\otimes T^\ast M)$ satisfying Gauss' equation \begin{eqnarray}...
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### Question on how differential form as defined for subsets of $\mathbb R^n$ and integration on them in Spivak, Calculus on Manifolds

If $V$ is a vector space, denote by $\Lambda^k(V)$ the space of alternating multilinear maps from $V^k$ to $\mathbb R$, i.e. the space of alternating $k$-tensors. Also for a point $p \in \mathbb R^n$ ...
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### Locally disk-preserving charts?

This is slightly vague as I've not yet come to terms with what I'm actually looking for. On $S^2$ we may choose charts (stereographic projection) such that the image of a disk (i.e. all points ...
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### Chain rule over manifolds

Let $X,Y$ and $Z$ be three manifolds, and let $F:X\to Y,G:Y\to Z$ be smooth functions. Fix a point $x\in X$. Prove that $$D(G\circ F)\mid_{x}=DG\mid_{F(x)}\circ DF\mid_{x}$$ By using coordinate ...
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### If $\delta c = 0$, does it follow that $d\xi = 0$?

Let $\mathcal{U} = \{\mathcal{U}_\alpha\}_{\alpha = \infty}^\infty$ be a locally finite open covering of the manifold $M^n$, with smooth functions $\lambda_\alpha$, compactly supported in $U_\alpha$. ...
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### A question on an exercise to show that unit sphere could not be covered by a single chart

The following is an exercise from Marsden et al, Manifolds, Tensor Analysis, and Applications in the first chapter on manifolds. First let me cite three essential definitions: Definition 1: Let $S$...
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### Munkres Analysis on Manifolds Differentiation Question

Below is a problem from Munkre's Analysis on Manifolds book. I'm unsure of how to approach this; it seems to me to apply the defintion of the derivitative, but I cannot seem to get that to work out. ...
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### signature of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ (using homology, cohomology)

I want to prove that the signature $\operatorname{sig}(M)$ of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ is nonzero. First of all, $M$ is a compact 24-dimensional manifold ...
I have read that given a smooth even dimensional manifold $M$ with an almost complex structure $J$, then $M$ is orientable and there is a canonical choice of orientation. Why is this the case? How ...