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

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90 views

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic ? Why?

Which of principal curvature, Gaussian curvature and mean curvature is intrinsic and why? Which book will have this explanation? I read a book (Remannian Geometry, Do Carmo, p129). It has a saying: ...
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1answer
141 views

Another differential topology lemma

Another lemma (1) Why can we assume $z=f(z)=0$ and that $U$ is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ...
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1answer
83 views

Proving that something is a manifold

I'm a beginner at differential geometry and I'm having some trouble with the following problem: Let $M \subset \mathbb{R}^n$ be a $k$-dimensional smooth manifold (smoothly embedded in ...
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1answer
32 views

Cont. Function smooth iff composition with submanifold inclusion is smooth

I'm trying to proof the following: Let $X$ be a smooth manifold, $X_0$ an open subset of $X$, $i: X_0 \to X$ the canonical inclusion, $Y$ another smooth manifold and $f: Y\to X_0$ continuous, then ...
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1answer
49 views

Diffeomorphic connected hypersurfaces

Given a four dimensional Lorentzian manifold $\mathcal{M}$ (a manifold with a metric $g_{\mu\nu}$ in the tangent bundle with signature (-1, 1, 1, 1)), we define a global spatial foliation by a ...
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1answer
68 views

Linear Subspace of $\mathbb{R}^n$ is a Manifold

How does one prove that a that a linear subspace of $\mathbb{R}^n$ is a manifold? This question arises from Spivak's Calculus on Manifolds, Chapter 5, problem 5-5: Prove that a k-dimensional ...
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1answer
45 views

For which values of $a$ is this set a manifold?

Let $f:\mathbb{R}^3\to\mathbb{R}, f(x,y,z)=(x-y+z-1)^2$. For which values of $a$ is $\{(x,y,z)\in\mathbb{R}^3:f(x,y,z)=a\}$ a 2-manifold? Instead of $(x-y+z-1)^2=a$ seems a better idea to write ...
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1answer
39 views

Prove that inverse of $f$ defines a manifold

Let $f:\mathbb{R}^3\to\mathbb{R}$ be given by $f(x,y,z)=z^2$. Prove that $0$ is not a regular point but $f^{-1}(\{0\})$ is a manifold. I divided this in two parts: $(1)\; 0$ is not a regular ...
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171 views

A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
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1answer
111 views

Trying to prove M is a manifold

Let $M$ be the set of all points $(x, y, z) \in \mathbb{R^3}$ satisfying both of the equations $x^3 + y^3 + z^3 = 1$ and $x + y + z = 1$. Prove that M is a manifold, except perhaps near the points ...
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32 views

Charts on a Manifold

Let $f^{1},\cdots ,f^r,\Phi ^1,\cdots , \Phi ^{n-r}$ be functions of class $C^{(1)}$ on an open set $D$. suppose that $F=(f^1\mid S,\cdots ,f^r\mid S)$ is a coordinate system for $S$, that $S=\lbrace ...
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35 views

existance of loop with finitely many point of intersection

for every loop on compact orientable surface exists freely homotopic loop with finitely many points of intersection. I see that it have to be true, but I can't prove it. I know Thom's theorem, Sard's ...
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55 views

Tangent space of all real symmetric matrices with a fixed rank

Suppose $M_r$ is the set of all real symmetric matrices of order $n$ with rank $r$. (a) Show that $M_r$ is a submanifold of the space $\mathbb R^{n^2}$. (b) Find the tangent space of $M_r$ at some ...
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1answer
59 views

Integral Curves of a Vector Field

How do I find the integral curves of a vector field and what are they intuitively? eg. what are the integral curves of vector field $X=\frac{1}{x}\frac{\partial}{\partial ...
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1answer
77 views

The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold

Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla ...
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30 views

Basic Manifold Question

Below is a paragraph from the appendix from Krantz's Several Complex Variables book. I have limited knowledge regarding manifolds and was hoping (very much) that someone would be willing to provide ...
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96 views

Why is surgery along a framed link well defined?

Let $L=L_1 \cup L_2 \cup \cdots \cup L_n $ be a framed link in $ S^3 $. I want to perform the surgery along $L$ to get a new manifold $M$. By definition, to perform this surgery, I must perform the ...
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161 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
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1answer
48 views

Alexander duality formulation + Jordan-Brouwer separation

In Davis & Kirk LNAT p.71 there is written: (1) How does this imply the Alexander duality $\tilde{H}^k(A)\cong \tilde{H}_{n-k-1}(\mathbb{S}^n\!\setminus\!A)$? (2) Is it assumed that the ...
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3answers
62 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
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39 views

Can the second order frame bundle of a 2-d manifold be embeded inside the ordinary frame bundle of a 3-d manifold?

I would like to know whether the second order frame bundle of a 2-d manifold can be embedded inside the ordinary frame bundle of a 3-d manifold? Explanation Suppose we have a 3D smooth manifold ...
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1answer
56 views

When can you recover a connection from totally geodesic submanifolds?

Let $g_{ab}$ a Riemaniann ( Lorentzian ) metric in a $n-$dimensional manifold $N$ and let $M$ be a submanifold of $N$. In general, the Levi-Civitta connection induced by the induced metric in $M$ ...
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45 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
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34 views

General linear group on manifold

Knowing that $\mathrm{O}(n,\mathbb{R})$ is a closed submanifold (of the general linear group) and that $\mathrm{SO}(n,\mathbb{R})$ is one of its subgroups with the same dimension, is there a quick way ...
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1answer
60 views

How to prove that a level set is not a submanifold

I've seen an exercise in which I am to prove that $$\{(x,y,z) \in \mathbb{R}^3 : x^2+y^2=z^2\}$$ is not a submanifold of $\mathbb{R}^3$. I've done a little bit of research and found these answers ...
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2answers
64 views

Need help finding Jacobian matrix of diffeomorphism of spheres

Let $S_a \subset \mathbb{R}^{n+1}$ and $S_b \subset \mathbb{R}^{n+1}$ be two spheres of radius $a$ and $b$ respectively. So $S_a$ are $n$-dimensional. Let $F:S_a \to S_b$ be the diffeomorphism $F(s) ...
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1answer
62 views

manifolds without symplicial or cell structure

In many situations in topology, (like the poincare duality) they put a distinction between the space being a manifold or just a cell or simplicial complex. I want to know why this is important, in ...
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60 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
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1answer
81 views

Van Kampen theorem on n-manifold remove one point

I'm trying to prove for a ($n \geq 3$)-dimensional path connected manifold M, if remove a point in M, then use Van Kampen theorem can somehow show the fundamental group of original $M$ and $M-{x}$, ...
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114 views

Is there any embedding theorem for fibre bundles?

I would like to know whether there is an embedding theorem for fibre bundles, like Whitney embedding theorem. When can a given fibre bundle be a subbundle of some higher dimensional bundle?
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66 views

Green Identities via Differential Forms

What is the actual meaning of the Green identities: Is there a picture/geometric interpretation of these, as well as intuition going beyond the usual integration-by-parts meaning associated to ...
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36 views

dimension of tangent space to a boundary point of a convex shape

I have a basic question regarding the dimension of the tangent space at a point $P\neq0$ that lies on the boundary of a pointed convex cone with its point centered at 0. For a 3D cone that is ...
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2answers
72 views

smooth surjective map from lower dimensional manifold onto higher dimensional manifold?

I'm thinking about whether there exists a smooth surjective map $f: M^m \twoheadrightarrow N^n$ where both $M$ and $N$ are smooth manifolds, and $\dim M = m < n = \dim N$. (We might assume that $M$ ...
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37 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
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1answer
39 views

Explicit Orientation-Reversing Homemorphism of $M_g$

Let $M_g$ be the orientable closed surface of genus $g$. I know that there is an orienation-reversing homeomorphism ($[M] \rightarrow -[M]$, where $[M]$ is fundamental class) $f:M_g \rightarrow M_g$ ...
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2answers
92 views

Geometry of Commuting Hermitian Matrices

I am a physicist working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of ...
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0answers
35 views

Asymptotic marginals of uniformly-distributed simplex

this is my first post. I am studying manifolds defined by linear transformations and noticed an interesting phenomenon I need to explain and don't know where to look. If I have an $N\times D$ maxtrix ...
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103 views

measurable function induces a measurable bundle

Greetings I am preparing a work on bundles and I found this statement Let $V$ a topological vector space with $\dim V=n$ and $(E,\pi,M)$ a vector bundle continuous over $M$ (compact space). If ...
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1answer
22 views

generation of sub bundle

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for ...
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41 views

Finding zeros of maps between manifolds with different dimensions

One of the questions for which the notion of degree is useful is: does this map have a zero. For example, one can prove the Fundamental Theorem of Algebra using the following fact involving the ...
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1answer
126 views

Orientability of submanifolds

What are some general conditions under which submanifolds of orientable manifolds will also be orientable. Of course this isn't true in general (for example, the Möbius band is a non-orientable ...
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2answers
120 views

lifting a product of commutators of standard generators on 2-manifolds

I have a problem with understand the proof http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf I don't understand this part: "(...) we can easily ...
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1answer
38 views

Simple closed curve definition of genus

The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it. Why must the simple closed curves be ...
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19 views

The time evolution of levi-Civita connection

Assume a smooth one-parameter family of Riemannian metrics $g_{t}$. Write $h:=\frac {\partial}{\partial t}g$. In addition, assume that the Levi-civita connection on the Riemannian manifold ...
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3answers
505 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
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1answer
81 views

Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
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1answer
23 views

Interacting Manifolds?

Usually, manifolds have certain internal properties which are being studied on their own. However, I was wondering if a field of mathematics exists where several separate manifolds are considered ...
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2answers
314 views

Why do folded concentric circles and rectangles form a hyperbolic paraboloid?

Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?
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1answer
118 views

Seeking a good book or site describing the 3-sphere

would you be able to recommend a good book/chapter or a web site on visualization / structural elements / projection of the 3-sphere. I am trying to locate a good information source on this subject, ...
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let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...