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

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3
votes
1answer
42 views

Collection of smooth real valued functions on smooth manifold has ring structure.

For any smooth manifold $M$, how do I see that the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and that every point $x \in M$ determines a ...
7
votes
1answer
67 views

How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of tangent $2$-planes? [duplicate]

A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a subbundle of dimension $k$. How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of ...
8
votes
2answers
67 views

Diffeomorphism between $\mathbb{P}^n$ and the submanifold of $\mathbb{R}^{(n+1)^2}$ consisting of certain matrices?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinates space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$ by $q(x) = \mathbb{R}x ...
2
votes
1answer
144 views

Total confusion about differential one-forms and non-coordinate bases

I asked this question recently (Basis of differential one-form confusion), thought I understood the answer, but now realise I don't. Lee (Introduction to Smooth Manifolds) says that at a point $p$ ...
1
vote
1answer
41 views

What is the relationship between the unit simplex and the nonnegative orthant?

I came upon this figure while reading something online. Pictured above is the intersection of the unit sphere in the nonnegative orthant and the unit simplex. Question: What is the relationship ...
2
votes
1answer
38 views

Optimization on manifold via Lagrange multipliers

Let the manifold $S$ in $\mathbb R^n$ be defined by $g(x)=0$. If $p$ is a point not on $S$, and $q$ is the point of $S$ which is closest to $p$, show that the line from $p$ to $q$ is perpendicular to ...
4
votes
1answer
31 views

If $M$ is Riemannian, then $\kappa_f \oplus f^*TN \cong TM$, where $\kappa_f$ is built out of kernels of the $Df_x$?

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. I know how to construct a vector bundle $\kappa_f$ built out of the ...
6
votes
2answers
246 views

Unique factorization of manifolds?

I wonder if there is a result on the unique factorization of manifolds. Call a topological manifold to be indecomposable if it is not homeomorphic to a product of manifolds of positive dimension. Is ...
4
votes
1answer
38 views

Constructing a vector bundle built out of kernels of the Jacobian?

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. How do I construct a vector bundle $\kappa_f$ built out of the kernels of ...
0
votes
0answers
22 views

Does every smooth manifold admit a smooth $\Delta$-complex structure?

I know that every smooth manifold $M$ admits a triangulation. That means, there exists some simplicial complex $K$ homeomorphic to $M$. Does this mean that it also admits a smooth $\Delta$-complex ...
1
vote
0answers
30 views

Why do those terms vanish if the metric is Hermitian?

On this [page][1], the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor ...
1
vote
2answers
28 views

Find dimension and tangent space of manifold $M = \{(p_1,p_2,p_3)\in \mathbb{R^3}: p_1^2-p_2^2+p_1p_3-2p_2p_3=0, 2p_1-p_2+p_3=3\}$

Given set $M = \{(p_1,p_2,p_3)\in \mathbb{R^3}: p_1^2-p_2^2+p_1p_3-2p_2p_3=0, 2p_1-p_2+p_3=3\}$. Show that M is smooth embedded manifold in $\mathbb{R}^3$ of dimension 1 and compute tangent space ...
3
votes
2answers
106 views

Introductory book on differential geometry for engineering major

I am an engineering major and looking for a straightforward, easy to understand basic book on differential geometry to get started. At starting point, I am not looking for a comprehensive book (may be ...
0
votes
0answers
17 views

Find a direct basis for boundary orientation

Let M in R4 be a manifold defined by equation d= $x^2$ + $y^2$ +$z^2$ and oriented by sgn $dx_1$^$dx_2$^$dx_3$. Consider the subset where d<=1. Show that it is a piece with boundary. Let x be a ...
0
votes
0answers
15 views

Positive reach..

Let $M$ be a smooth closed $d$-manifold embedded in $\mathbb{R}^n$, and let $d_M: \mathbb{R}^n\to \mathbb{R}$ be given by $d_M(x) = \min_{m\in M} d(x,m)$. Define $M\oplus \epsilon = \{ x\in ...
0
votes
1answer
23 views

Why the function $f$ is $\mathcal C^\infty (M)$?

Let $M$ a manifold of dimension $n$. Let $g_\varepsilon:\mathbb R^n\longrightarrow \mathbb R$ s.t. $g_\varepsilon\in\mathcal C^\infty (\mathbb R^n)$ and ...
1
vote
1answer
24 views

Prove that $T_pM^*=\mathfrak m_p/\mathfrak m_p^2$.

Let $M$ a manifold of dimension $n$ and $T_pM$ its tangent space. Let denote $T_p^*M=\mathcal L(T_pM,\mathbb R)$ it's dual. Let also denote $$\mathfrak m_p=\{f\in\mathcal C^\infty (M)\mid ...
18
votes
3answers
338 views

Is the tangent bundle of $S^2 \times S^1$ trivial or not?

As the question title suggests, is the tangent bundle of $S^2 \times S^1$ trivial or not? Progress: I suspect yes. If I could construct three independent vector fields, I would be done. But I'm not ...
2
votes
0answers
74 views

Correct definition of gradient (and divergence) on smooth manifolds (for engineers)

I am very sorry if this is a very trivial question or my formulation is inadequate, I am only an engineer, and I am not familiar with the topic. I am looking for the definition of the gradient of a ...
2
votes
2answers
37 views

Mod p cohomology ring of real projective space

I have looked at five algebraic topology books, but the only computation of cohomology ring of $\mathbb{RP}^\infty$ which I could found is with $\mathbb{Z}_2$-coefficient : $$ ...
1
vote
1answer
22 views

Esitimate for boundary points and exterior normal from Jost's Partial Differential Equation 3rd edition

This is a quetion I met at Jost's Partial Differential Eqution 3rd edition. In p105, an inequality that is used to prove Lemma 5.3.1 . Here is the statement subtracted from its original argument. Let ...
0
votes
1answer
25 views

Manifold and derivative: change of coordinate.

I'm stuck on the following problem. Let $M$ a manifold of dimension $n$ and let $\varphi_1:U_1\longrightarrow W_2$ and $\varphi_2:U_2\longrightarrow W_2$ two charts at the neighborhood of $p\in ...
1
vote
0answers
16 views

Does time-reversal turn the stable manifold of an equilibrium into unstable manifold?

If a trajectory $x(t)$ starts in the stable manifold of an equilibrium, it tends to the equilibrium as $t\to +\infty$. If we now consider the opposite direction of $t$ (i.e. draw the arrow that marks ...
11
votes
2answers
229 views

What are examples of parallelizable complex projective varieties?

A smooth complex projective variety is the zero-locus, inside some $\mathbb{CP}^n$, of some family of homogeneous polynomials in $n+1$ variables satisfying a certain number of conditions that I won't ...
0
votes
1answer
12 views

Let $W$ a neighborhood of $p\in M$. Why is there $V\subset M$ s.t. $V\subset \overline{V}\subset W$?

Let $M$ be a manifold and let $W$ a neighborhood of $p$. Why is there an open $V$ s.t. $p\in V\subset \overline{V}\subset W\subset M$ where $V$ is compact ? It's in a proof of a theorem of my course, ...
7
votes
1answer
52 views

Boy's surface, visualization of the preimage of self-intersection locus as graph on projective plane

For the immersion of the projective plane in $\mathbb{R}^3$ with one triple point, what does the preimage of the self-interaction locus as a graph on a projective plane look like?
3
votes
1answer
59 views

Construction of an immersion of $T^3$ − point in $\mathbb{R}^3$?

Let $p\in T^3$. How do I construct an immersion of $T^3\setminus\{p\}$ in $\mathbb{R}^3$?
1
vote
0answers
25 views

Confusion about the definition of Sobolev spaces on manifolds

Let $(M,g)$ be a manifold with metric $g$ parametrized by the mapping $S$ and parametric domain $\Omega$. The sobolev space of order one with respect to the $L_2(M)$-norm $H^1_2(M)$ is defined as ...
2
votes
2answers
37 views

Constructing a (smooth) diffeomorphism between non-smooth manifolds

I've been trying to construct a smooth diffeomorphism between non-smooth manifolds. Unfortunately I don't think I know enough manifolds well enough to find an example of this. Mostly I've been ...
2
votes
0answers
27 views

Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold

I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem: Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...
0
votes
0answers
12 views

Approximations for an asymptotically flat manifold.

Let us consider an asymptotically flat manifold $M$ with the metric $g$. Let $\delta$ denote the flat metric and $p = g- \delta$. I am given the following approximations, where $C$ is a constant. 1) ...
1
vote
1answer
19 views

Open neighborhood of a manifold boundary point

Manifold with boundary: An $n$-dimensional manifold with boundary is a second countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset of ...
8
votes
2answers
229 views

Do the singular matrices form a topological manifold

So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$). I have two related questions: Is the set of ...
1
vote
0answers
73 views

Show solution is an attracting center manifold.

Consider the small solutions of the following system \begin{align} \dot{x}&=\epsilon x-x^3+xy \\ \dot{y}&=-y+y^2-x^2 \\ \dot{\epsilon}&=0 \end{align} with $0<\epsilon\ll 1$. Show ...
1
vote
1answer
24 views

In finding boundary of the product of two half-lines, shall homeomorphism be global?

Lets $ \mathbb{R_{+}^{n}} = \mathbb{R^{n-1}} \times [0;+\infty[ $ Basically in my course I have this statement within the definition of a manifold with boundary: $ \forall x \in M, \exists U_x $ an ...
2
votes
1answer
42 views

Manifolds as Homogeneous Spaces

With very little effort one can, for example, show that $S^n$ can be written as a homogeneous space as $S^n\cong G/H$, where $G$ is the group of all rotations in $\mathbb{R}^{n+1}$ about the origin ...
1
vote
1answer
25 views

Reference for a result

A friend of mine told me that the cohomology of $\pi_1(M)$ was isomorphic to the cohomology of the manifold $M$. Is that true (maybe there are some hypothesis) ? Does someone know a reference for this ...
7
votes
1answer
44 views

Tangent manifold $D(M_1 \times M_2)$ is canonically diffeomorphic to the product $DM_1 \times DM_2$.

Let $M \subset \mathbb{R}^A$ and $M_2 \subset \mathbb{R}^B$ be smooth manifolds. How do I see that the tangent manifold $D(M_1 \times M_2)$ is canonically diffeomorphic to the product $DM_1 \times ...
2
votes
1answer
58 views

Differential forms, projections

I have a problem with one exercise from differential geometry. I don't even know how to start. Anyone could help with this problem? Let $M$, $N$ be manifolds, $M$ connected. Let $\pi:M\times N \to N ...
0
votes
1answer
17 views

Divergence, contraction and lie derivatives

I'm working through this question. I can show the forward direction in (a) but can't show the converse. I have $\delta/\delta t \phi^*_t \mu$ evaluated at t=0 is 0, but I can't see how I conclude ...
1
vote
1answer
35 views

Clarification about differentiable manifolds

When checking the transition maps for differentiability in order to determine if a manifold is differentiable, do we fill in any removable singularities (i.e. simplify the function composition before ...
2
votes
2answers
30 views

Lie derivative of the product of a function and a form

If $V$ is a vector field and $\alpha$ is a linear form, $$ L_V\alpha: X\mapsto V(\alpha(X))-\alpha([V,X]) $$ for every field $X$, is $\mathbb R$-linear, which is trivial since $V$ is a field and ...
0
votes
0answers
38 views

Rigorous Definition On composition of Multivariable Functions

Suppose you have $f(x_1,.....x_n)\colon R^n \to R$ and $g_1,...g_n\colon R^m \to R$. Then $f(g_1,....g_n)$=$f\circ(g_1,...g_n)$ :$R^m \to R$, right? Any clear definition on composition of ...
0
votes
1answer
44 views

What's the cohomology of disjoint union of two circles

I am computing the cohomology of $T^2$ by Meyer-Vietoris sequence. $T^2$ can be seen as the union of two open sets U and V s.t. U and V are diffeomorphic to a cylinder respectively. Thus U$\cap$V is a ...
2
votes
1answer
64 views

Riemannian manifolds isometry

Here is the following problem: Let $g_0$ be the Euclidean metric on $\mathbb C=\mathbb R^2$. Let $M=\{z \in \mathbb C| \ |z|<1 \}$ and equip it with the Riemannian metric ...
5
votes
1answer
181 views

Non-diffeomorphic structures on the sphere

How many smooth structures are there on $S^2$, $S^3$, and $S^4$ up to diffeomorphism? I looked around and couldn't find an answer; two books I have say different things on the subject. I know one of ...
6
votes
1answer
57 views

A product of smooth manifolds together with one smooth manifold with boundary is a smooth manifold with boundary

Suppose $M_1, \dots M_k$ are smooth manifolds and $N$ is a smooth manifold with boundary. Then how do I see that $M_1 \times \dots \times M_k \times N$ is a smooth manifold with boundary, ...
2
votes
1answer
31 views

Using Mayer-Vietoris to show $\chi(M) = \chi(U)+\chi(V)-\chi(U \cap V)$

Let $M$ be a manifold, and $U$, $V$ open sub-manifolds in $M$. How would one use the Mayer-Vietoris theorem to show that $\chi(M) = \chi(U)+\chi(V)-\chi(U \cap V)$, where $\chi$ is the Euler ...
0
votes
2answers
36 views

Manifold and maximal atlas

1) I didn't understand really what is a maximal atlas. Is it as set of compatible chart maximal in the sens that adding one more chart will yield the atlas not compatible ? 2) Let two atlas $\mathcal ...
3
votes
1answer
26 views

Why is it impossible for a compact connected non-orientable $n$-manifold to be the suspension of some connected based space?

Suppose $M$ is a compact $n$-manifold (without boundary) that equals the reduced suspension $\Sigma Y$ of a connected based space $Y$. Why must $M$ be orientable? I am aware that cup products vanish ...