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

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Norm Inequality on a Compact Riemannian Manifold

Consider the following problem: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality $$\Delta u \geq ...
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1answer
18 views

Proof formalization help: Given a vector $u$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at 2 points.

Proof formalization help: Given a vector $u$ of Euclidean length $1$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at at least 2 points. I've thought about the ...
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0answers
13 views

covering number of cartesian product of manifolds

Let there be two manifolds $P$ and $Q$, with their covering numbers $N_P(\rho)$ and $N_Q(\rho$), respectively. Is it true that the covering number for the Cartesian Product $P\times Q$, $N_{P\times ...
3
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1answer
109 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
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0answers
46 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
4
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1answer
151 views

quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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37 views

Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
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1answer
28 views

Under which additional hypothesis are open maps locally injective

Recollection of basic definitions: We recall the basic definitions that a continous map of topological spaces $f : X \to Y$ is open if $f(U)$ is an open subset of $Y$ whenever $U$ is an open subset ...
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2answers
41 views

Showing that the set of semi-orthogonal matrices is a $C^\infty$ submanifold

For $k, n \in \mathbb{N}$ with $k ≤ n$, we define $$S_{n, k} = \{X \in \mathbb{R}^{n \times k}: X^t X = I_k\}$$ where $I_k$ is the identity matrix of rank $k$. I want to prove that $S_{n, k}$ is a ...
3
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1answer
67 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
2
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1answer
287 views

“Completing” a vector field on a non-compact manifold $M$

Suppose I have a non-compact smooth manifold $M$ and an arbitrary nowhere-vanishing smooth vector field $X$ on $M$ which is not complete. Is there a way to create a smooth vector field $V$ that is ...
2
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1answer
50 views

Is the cuspidal cubic $\{y^2 = x^3\} \subset \Bbb R^2$ not smooth?

Cuspidal cubic $y^2=x^3$ in $\Bbb R^2$ "seems to be not smooth" intuitively because its pictured graph has a cusp at the origin. But I read from book that it is a smooth manifold. I feel so confused. ...
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0answers
11 views

explaination of the metric tensor on another manifold?

In skew -product decomposition the following features are observed :- 1.the Riemannian Manifold $(M,g)$ has a product form of $$M=R\times \Theta$$ Where $\Theta ,R $ are connected $C^\infty$ ...
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0answers
22 views

the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
3
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0answers
31 views

any open set in $\mathbb{R}^n$ is a $n$ dimensional manifold

I am trying to show this using the definition: M is a k-dimsensional submanifold of $\mathbb{R^n}$ if for all $x \in M$ the following condition holds: There exists an open set $U \subset ...
2
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2answers
63 views

What does $C^k$ at a single point mean?

I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean? Furthermore if $f$ is $C^k$ at $p$, is it ...
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26 views
0
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31 views

Manfolds, Tangent Space, Cotangent Space, forms, adding vectors [on hold]

Could someone explain in WORDS, what is going on with these words in the Title. Why are they important? I am teaching myself a bit about all of them. I have the general idea. But I am looking for ...
1
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1answer
75 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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0answers
20 views

What is the motivation for wanting to remove redundant terms that arise from the wedge product of two multilinear functions?

I am currently working through Loring Tu's "An Introduction to Manifolds," specifically the section in which the exterior algebra of multicovectors is introduced, and I am having trouble grasping the ...
5
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1answer
588 views

Local homology group: a homeomorphism takes the boundary to the boundary

Let $X \subset \mathbb{R}^{n}$ be the subspace $\{ x_{1},...,x_{n} \mid x_{n}\geq 0 \}$, and let $Y$ is the subspace with $x_{n}=0$. Let $x\in Y$, calculate the local homology of $X$ at ...
4
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1answer
35 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
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1answer
42 views

Two proof of Petersen's 'Manifold'

Picture below is from the 5 page of Petersen's Manifold. First, why diffeomorphism is forced to be a lieanr isomorphism? The define of diffeomorphism accords to Wiki. Second , what space the point ...
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0answers
19 views

Why is $H_{DR}^p(M,\mathbb{C})\cong H_{DR}^p(M,\mathbb{R})\otimes_\mathbb{R}\mathbb{C}$

This question is related to my previous question. The answers to that question inspired a new question, namely For a complex manifold $M$, why is $H_{DR}^p(M,\mathbb{C})\cong ...
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1answer
41 views

Poincaré Duality in Middle Dimension

I am reading a paper that states the following theorem without proof: Poincaré duality in middle dimension: Let $M$ be a connected oriented manifold of even dimension $2d$. Then the cup product ...
4
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1answer
67 views

Why is the image of the implicit function in the implicit function theorem not open?

We have a continuously differentiable function $f$ from $\mathbb{R}^{n+m}$ to $\mathbb{R}^n$, and we find a continuously differentiable function $g$ which maps points from $\mathbb{R}^m$ into ...
6
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2answers
831 views

Product of manifolds & orientability

I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable. I am able to prove that the product is ...
4
votes
1answer
486 views

How to find the Frechet derivative at $A\rightarrow A^{-1}$ mapping?

I am reading on my own the Lectures on the Geometry of Manifolds (http://nd.edu/~lnicolae/Lectures.pdf ) , and got stuck in solving the exercise 1.1.3 (b) . The 1.1.3 (b) is : Let F: $U\rightarrow ...
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1answer
32 views

Left-invariant vector fields on the circle $S^1$

I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...
0
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1answer
25 views

On computing the Differential of a Smooth Map

In class, we proved Jacobi's formula for the differential of the determinant using the following formula for the differential of a smooth map $F$ between manifolds $M$ and $N$ $$D_A F(B) = ...
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0answers
24 views

Showing that a set is a smooth submanifold

How do I show that the solution set to the equations $x^3+y^3+z^3=1$ and $z=xy$ is a smooth manifold? Regular value theorem doesn't apply here and I don't know how to construct an atlas for it.
2
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1answer
31 views

Why is $\phi^* g = g$ a PDE for a pseudo-Riemannian metric $g$ on a manifold?

Given a (locally trivial) bundle $\pi: E \to M$ a PDE of order $k$ is usually defined to be a submanifold of the jet-bundel $J^k(E)$. Now assume $E = M \times M$ and $\pi$ is the projection on the ...
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1answer
40 views

Derivative of map $f: S^n \to \mathbb{R}P^n$ is an isomorphism

I'm trying to show that the map $f: S^n \to \mathbb{R}P^n$ given by sending a unit vector $x$ in $S^n \subset \mathbb{R}^{n+1}$ to the line spanned by $x$ in $\mathbb{R}P^n$ has injective derivative. ...
0
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1answer
43 views

Why is a metric?

I have a question about tensors and metrics: Let $M=\{(t,x,y,z)\in \mathbb{R}^4: t>-1 \}$ and let $g=(1+t)dtdx+dy^2+dz^2$ Show that g is a metric on $M$. I did the next, I have the basis $\{ ...
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0answers
68 views

Which homology groups of a closed orientable 6-manifold can be isomorphic to $\mathbb{Z}^3$?

List all $i$ for which there is a closed orientable $6$-manifold $M$ with $H_i(M) =\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ I am working on an old exam problem and this one stumped me. ...
0
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1answer
32 views

Difference between Grassmann and Projective space?

I recently read the first few chapters of Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and I am confused about ...
0
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0answers
17 views

Quadrature over a (smooth, compact, convex, etc.) manifold

Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ...
2
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1answer
40 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
2
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2answers
51 views

Representable homology classes on smooth manifolds

Let $X$ be a closed (compact without boundary) smooth manifold. We can consider its singular homology $H_*(X,\mathbb{Z})$. Let $H_{k}(X,\mathbb{Z})$ be the $k$-th singular homology group of $X$ and ...
2
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1answer
53 views

Question about connections on the dual bundle.

Let $E \to M$ be a vector bundle with connection $\nabla$. Extend $\nabla$ to $E^*$ and $E^* \otimes E$ in the regular fashion. Is $\text{Id} \in E^* \otimes E$ necessarily parallel?
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32 views

When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
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1answer
23 views

Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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1answer
229 views

Orientability of Grassmannians

How can I understand when $Gr(n,k)$ is orientable and when not? I found that answer is yes if and only if $n \vdots 2$, but I do not know how to prove it.
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1answer
47 views

Every $\mathcal{C}^1$ manifold can be made smooth?

I heard of a theorem saying that each $\mathcal{C}^k$-manifold with $k\geq 1$ can be made into a smooth manifold, i.e. $\mathcal{C}^{\infty}$ (by restriction of the atlas). However, I cannot find ...
2
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1answer
27 views

Lemma characterizing second fundamental form, do not understand step

Consider an excerpt of a lemma and part of its proof from a Riemannian geometry textbook. Lemma. The second fundamental form is independent of the extensions of $X$ and $Y$; bilinear ...
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1answer
58 views

Are the collections {rings}, {smooth manifolds} larger or smaller than {ab. groups}, {top. manifolds}?

Intuitively, the collection of smooth manifolds feels smaller than that of topological manifolds: they are not just locally nice continuous objects, but even smooth. Similarly, when one finds that an ...
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1answer
38 views

Quotient by a discrete subgroup of a Lie group

I was reading Fulton Harris' Representation theory, A first course, where I came across the following: Let $H$ be a Lie group and $T$ be a discrete subgroup of its center $Z(H)$. Then there exists ...
3
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1answer
27 views

Understanding Hempel's proof of uniqueness of cube with handles

In Hempel's 3-Manifolds book, Theorem 2.2 says that if $P$ and $Q$ are two cubes, both with $n$ handles, and both are orientable, then they are homeomorphic. He defines a cube with handles as a ...
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1answer
25 views

Is every convex cone a manifold?

Let $C \subseteq \mathbb{R}^n\setminus \{0 \}$ be a connected convex cone*. Question: Is $C$ always a topological manifold (perhaps with boundary)? A smooth one? Does anything change if we do not ...
0
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1answer
30 views

Vector bundle vs Total Space

On page 59 in Lee's "An Introduction to Smooth Manifolds" the author writes, "Let $E$ be a smooth vector bundle over a smooth manifold $M$, with projection $\pi:E\to M$." I thought the vector bundle ...