Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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18 views

Vanishing pushforward implies smooth function is locally constant?

I'm trying to prove that if the pushforward $dF$ of a smooth map $F\colon M\to N$ between smooth manifolds is zero, then $F$ is constant on each component. It will be enough to show $F$ is locally ...
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1answer
11 views

Inverse Image of a Regular Value an Orientable Submanifold

Let $f:M^n \rightarrow \mathbb{R}$ be a smooth map, and let $c\in N$ be a regular value. When is $f^{-1}(c)$ an orientable manifold? Note: I know by regular value thm, $f^{-1}(c)$ is a smooth $n-1$ ...
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1answer
25 views

Orthonormal basis for a tangent plane

Given a manifold $M$ described by the graph of an arbitrary smooth function $f:U \subset \mathbb{R}^2 \to \mathbb{R}^3$, I would like to construct an orthonormal basis for its tangent plane $T_pM$ at ...
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0answers
29 views

there exists a unique plane in a point of a surface in $\mathbb{R}^3$ [on hold]

The question is how I can prove the existence in this problem: If $M\subset \mathbb{R}^3 $ is a smooth surface. Then, there exists a unique plane $\Gamma\subset \mathbb{R}^3$ that passes through ...
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0answers
19 views

Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
2
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3answers
216 views

Holomorphic functions on a complex compact manifold are only constants

Is there a simple proof that every holomorphic function $M\to\mathbb{C}$ on a compact complex manifold $M$ is constant?
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1answer
21 views

Estimate for boundary points and exterior normal vector of bounded domain of class $C^2$

Consider a bounded open set $\Omega\subset\mathbb{R}^d$, s.t. the boundary set $\partial \Omega$ is a manifold of class $C^2$. Let $x,x_0\in\partial\Omega$ be boundary points and $\nu_x$ the exterior ...
3
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0answers
27 views

Doubt with smooth extensions

Let $(x,y)\in\mathbb R^2$ and $M$ a manifold defined by $M=\left\{ (x,y)\in\mathbb R^2\, |\, y^2+x=0 \right\}$. Let $\pi$ be a projection $\pi(x,y)=(x)$. Let $\phi:\mathbb R\to\mathbb R$ be a ...
0
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0answers
29 views

Parallel Transport of Geodesic Velocity Vectors

Given a Riemannian manifold $M$ with Riemannian metric $g_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ and distance $d:M\times M\rightarrow\mathbb{R}$ determined by length of minimizing geodesics, ...
3
votes
1answer
51 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
3
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0answers
45 views

Non-ellipticity of Yang-Mills equations

How does one prove that the Yang-Mills equations from classical Yang-Mills theory are not elliptic? Alternatively, how does one calculate the principal symbol of the Yang-Mills equations? Can ...
1
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1answer
32 views

Levi Civita connection along principal curvature directions

Let $(M,g)$ be a surface that can be immersed into $\mathbb{R}^3$. Denote by $\nabla$ the associated Levi Civita connection. Further, let $X_1,X_2$ be the directions of principal curvature which are ...
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0answers
30 views

Question on theorem: $\delta_{g}F(x)=0\Leftrightarrow\left[g,F\right]=0$ [on hold]

Can anyone explain me why the following statements are the same? $$\delta_{g}F(x)=0\Leftrightarrow\left[g,F\right]=0$$ Is there any theorem with this statement? Thanks in advance!
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24 views

Connection defined by its geodesics

I have a question related to the definition of a connection (in the sense of Koszul) by its geodesics. I know that a torsion free connection is uniquely determined by its geodesics. Now, let $M$ be a ...
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1answer
44 views

Is a level set of a manifold a set of zeroes

Suppose $X$ and $Y$ are manifolds of dimensions $k$ and $l$ (with $k>l$). Given $F : X \to Y$ a smooth map and $y$ a regular value in $Y$, does there exist a map $G : X \to \mathbb{R}^l$ such that ...
0
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1answer
15 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-29

I completed near all problems om a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example u = ...
1
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1answer
14 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-28

I completed near all problems of a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example $u = ...
1
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2answers
72 views

Non existence of a non singular vector field on $S^2$

Prove that the unit tangent bundle of $S^2$, $T^1 S^2$, is not diffeomorphic to $S^2×S^1$ by showing that if so there exists a nowhere vanishing vector field on $S^2$ I do not know how to create that ...
9
votes
2answers
132 views

Is there a retraction of a non-orientable manifold to its boundary?

It's easy to show using Stokes theorem that a compact orientable manifold with boundary cannot retract to its boundary, by choosing a volume form. But for the non-orientable case I don't know if this ...
3
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0answers
30 views

Second fundamental form of a graph of a function using frame fields

I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with ...
2
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0answers
33 views

Connections on principal bundles and vector bundles

In Donaldson and Kronheimer's book on the geometry of four manifolds, a brief review of connections on principal bundles is given. Three equivalent definition are stated: 1) Via horizontal subspaces, ...
4
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0answers
47 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
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0answers
19 views

Show differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ is injective

The problem is find to the differential of $f:S^{m}\times S^{n}\to S^{m+n+mn}$ (spheres) defined as $f(x_{0},...,x_{n},y_{1},...y_{n})=(x_{0}y_{0},x_{0}y_{1},...,x_{n}y_{1},x_{n}y_{n})$ and show it is ...
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1answer
24 views

Differential of rotation matrix at the north pole of sphere

Let T(p) rotate $p\in S^{2}$ by angle $\theta $ about the z-axis. The problem is to compute $dT_{(0,0,1)}$. T can be represented by the usual 3x3 rotation matrix $A_{z}(\theta)$. So $T(p)=A_{z}p$. ...
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0answers
23 views

Coordinate frames along the bounday of a minimal area (soap-film) surface

I would like to calculate coordinate frames along a closed Bezier (Or Catmull-Rom) spline. One axis should be tangential to the curve, and another axis normal to the minimal-area surface (soap-film ...
1
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1answer
27 views

Difference betwee parameterization and embedding of manifolds

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?
7
votes
2answers
112 views

Smooth surfaces that isn't the zero-set of $f(x,y,z)$

The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth ...
1
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3answers
59 views

May directed graph be embedded into manifold?

May directed graph be embedded into manifold?How ?and what is the condition?
2
votes
2answers
35 views

Calculus III: Find the points of the curve…

I have to find the points of the curve $$r\left( t \right) =\left( t,{ t }^{ 2 },{ t }^{ 3 } \right) $$ where the osculating plane passes through the point $\left( 2,-\frac { 1 }{ 3 } ,-6 \right)$.
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0answers
40 views

The name of this metric.

Does anyone know what metric is the following one in $\mathbb{R}^n$? $g=\sum_{i=1}^{n}(1+x_i^2)dx_i^2+2\sum_{i\neq j}x_i x_j dx_i dx_j$ Thanks in advance.
2
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0answers
18 views

Reference for Envelope, Evolute and involute

I have to give a lecture on Envelopes, Evolute and Involute to I year undergraduate students. Please suggest me some books which explain these concepts with examples geometrically. Already I have seen ...
0
votes
1answer
14 views

contraints on equation of a cylinder

(x-a)^2+(y-b)^2 = r^2 any way to adjust this formula to add constraints to the z-axis? recently introduced to the idea this goes forever in z-axis and I want to see if theres way's to adjust formula ...
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0answers
46 views

Zeros of vectorial field [on hold]

Given a $M$ manifold in ${\mathbb R}^n$ and $X:M\rightarrow TM$ a vectorial field such that $\pi\circ X=Id$ where $\pi:TM\rightarrow M$ (projection to $M$). One zero of $X$ is such that ...
6
votes
1answer
38 views

Flat connection with non-trivial holonomy? I cannot get it

maybe this is a dumb question, but I cannot understand how a principal $G$-bundle can have non-trivial holonomy with a flat connection. Maybe I'm missing something, but doesn't Ambrose-Singer theorem ...
1
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1answer
29 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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2answers
34 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
0
votes
0answers
24 views

Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
0
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1answer
30 views

calculate normal field of cylinder and sphere

in a book it is given that the unit normal field of $S^2$ is $N(p)=p$ the identity map. pictorially it is clear to me. But if I take any point on the sphere and multiply(usual scalar product of ...
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1answer
20 views

How to find these “breaking” points on an offseted curve?

Check the picture: I noticed that for big offset values the offseted curves often "breaks" like this (one or more times). So my question is that what is that point marked with ':-(', and how can I ...
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0answers
52 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
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0answers
41 views

Derivative with respect to a vector and tensor on a manifold

I am reading through a paper and have come across a statement which I do not fully understand. I paraphrase below. Consider a scalar function $f = ...
3
votes
2answers
90 views

Book for Undergrad Differential Geometry

I am soon going to start learning differential geometry on my own (I'm trying to learn the math behind General Relativity before I take it next year). I got the sense that a good, standard 1st book ...
3
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0answers
62 views

Submanifold of $2\times 2$ Complex Matrices Using Transversality?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
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0answers
19 views

Why is $\frac{d}{d \mu} \bigg|_{\mu=0} \, F^*_\mu F^*_\lambda t = F^*_\lambda \frac{d}{d \mu} \bigg|_{\mu=0} \, F^*_\mu t $?

In the book "Manifolds, Tensor Analysis, and Applications" by Marsden, Ratiu, Abraham the following relation (see the proof of 6.4.1, third edition) is used: $$\frac{d}{d \mu} \bigg|_{\mu=0} \, ...
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1answer
31 views

extension of a local orthonormal frame on a hypersurface

Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along ...
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0answers
28 views

Why are the charts in this set all $C^\infty$-related?

I'm reading the first volume of Spivak's differential geometry series, and am having a tough time convincing myself of something mentioned in the proof of Lemma 1, Chapter 2. Let $M$ be an ...
2
votes
1answer
68 views

relative sign in Hodge star of tensor product

Let $V$ be a vector space of arbitrary (finite) dimension and let $(V, \langle \ ,\ \rangle, I) = (W_1, \langle\ ,\ \rangle_1, I_1) \oplus (W_2, \langle\ ,\ \rangle_2, I_2)$ be a direct sum ...
0
votes
1answer
35 views

understanding the definition of a regular surface

Let $S\subset \mathbb{R}^3$ be a subset. We call $S$ a regular surface if there exists for every point $p \in S$ an open neighbourhood $V$ of $p$ in $\mathbb{R}^3$, and if, in addition, there exists ...
4
votes
2answers
39 views

Geometrically, what is the stereographic projection of a closed $n$-ball?

To show $\overline{B^n}$ is a $n$-manifold with boundary, apparently there is a trick to use stereographic projection after subtracting out the radius connecting $0$ to the north pole. I'm familiar ...
3
votes
0answers
66 views

Complex structures on Riemann surfaces

Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq H^0(M;K^2)$. Considering $\alpha$ as a map $T^{0,1} M \to T^{1,0} M$, the bundle $$ \{v + \alpha(v) \mid v \in T^{0,1} M\} ...