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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Is there any large diffeomorphisms of $S^{n}\times S^1 $like Torus?

We know that a Torus is mapped onto itself in a special discontinuous transformation given by $PSL(2,\mathbb{Z})$. Thinking of torus as $S^{1}\times S^{1}$ and thus as a lattice, we can easily show ...
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Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
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Maps between tangent space of product manifold and sum of tangent spaces

I am trying to prove that $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$ We define: $$\Phi:T_{(p,q)}(M\times N)\to T_pM\oplus T_qN:v\mapsto(d_{(p,q)}\pi_M v,d_{(p,q)}\pi_N v)$$ and $$\Psi:T_pM\oplus ...
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Exercise concerning areas inside closed curves

Let $\alpha (s)$, $s\in[0,l]$, be a closed, convex, plane curve with $\kappa >0$. Let $r$ be a positive constant and define $\beta (s)=\alpha (s)-rn(s)$, where $n(s)$ is the normal vector of ...
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Showing $K_1= H+\sqrt{H^2-K}$ and $k_2=H-\sqrt{H^2-K}$

I am trying to show the principle curvatures $k_1$ and $k_2$ are given by are given by $K_1= H+\sqrt{H^2-K}$ and $k_2=H-\sqrt{H^2-K}$ This is what I have so far: The Gaussian curvature $K$ and mean ...
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24 views

Construct a bump function for upper hemisphere

When reading section 13.1 of Loring Tu book, I came across this problem on constructing a bump function. Write down an explicit function $f : S^2 \to \mathbb{R}$ such that $f(p) = 1$ for all ...
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33 views

Why are scale factors not always unity?

A scale factor in curvilinear coordinates is defined as $$h_v \equiv \left|\frac{\partial\vec{r}}{\partial v}\right|$$ where $\vec{r}=(x,y,z)^T$ is a position vector. The partial differential can be ...
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27 views

Show $f$ is an isometry from $s$ to $s'$

Let $s$ denote the surface of revolution $$(x,y,z)=(\cos \theta \cosh v, \sin \theta \cosh v,v)$$ where $0 < \theta < 2 \pi$ and $-\infty < v < \infty$ Let $s'$ denote the surface ...
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15 views

Expressing the Christoffel symbols of the first and the second kind in terms of the metric tensor $g$ and expressing the symmetries

I am trying to expressing the Christoffel symbols of the first and the second kind in terms of the metric tensor $g=g_{ij}$ and express the symmetries of each with respect to the permutation of the ...
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27 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on ...
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45 views

Is this picture of the covariant derivative correct

I am reading O'Neil's Elementary Differential Geometry on my own. On page 81 he gave the following definition: Let $W$ be a vector field of $\mathbb{R}^3$, and let $v$ be a tangent vector field to ...
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20 views

Justify the identity $\frac{\partial}{\partial x^i} (\log|\det A|)=b_{rs} \frac{\partial a_{rs}}{\partial x^i}$

Let $A=[a_{ij}(x)]$ be a non singular matrix valued function with inverse $A^{-1}=B=[b_{ij}(x)]$ I am trying to use the chain rule to justify $\dfrac{\partial}{\partial x^i} (\log|\det ...
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Height function of a hypersurface

I was reading an article by do Carmo and Warner, which says: "By the height function for an oriented hypersurface at a point $p$ we shall mean the function defined on a neighborhood of the origin in ...
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14 views

divergence theorem with singularity at r = 0

I am trying to evaluate the volume integral given by \begin{align} \int_V [\nabla(\vec{x} \cdot \vec{u}) - \nabla \cdot (\vec{x}\vec{a})] dV \end{align} where $\vec{x}$ is the position vector and ...
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39 views

integral of a vector field in $\mathbb{R}^n$

I'm wondering the definition of the integral of a vector field on a hypersurface in R^n. Here is what I guess, but I did not found it on the internet. Let $v$ be a vector field on $\mathbb{R}^n$ and ...
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36 views

Sobolev spaces on Riemannian manifold

I know one can define the Sobolev spaces on a Riemannian manifold as completions, but is there an equivalent definition that uses weak derivatives, like in the case of open sets in $\mathbb{R}^n$ ? ...
4
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69 views

The tangent space of a vector space

I'm trying to show that there is a canonical isomorphism between a finite-dimensional vector space $V$ (regarded as a $C^\infty$ manifold) and its tangent space $T_vV, v\in V$, without using a basis, ...
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48 views

Whether there is easy way to compute $R_{ij}=\frac{1}{2}Rg_{ij}$ in 2-dimension

In 2-dimensional Riemann manifold ,Ricci curvature is given by $$ R_{ij}=\frac{1}{2}Rg_{ij} $$ My PDE teachers teach me to compute it by the way. $$ R_{11}=g^{ij}R_{1i1j}=g^{22}R_{1212} \\ ...
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43 views

Is there a Mobius (infinite) cylinder?

In order to understand the question of the title I need to understand another thing first. If we consider the Mobius band, locally, for a $U_i \subset S^1$, where $S^1$ is the base space, the bundle ...
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27 views

Riemann Curvature tensor for surfaces

Let $M$ be a regular surface on $\mathbb{R}^3$. I am trying to express the Riemann's curvature tensor: $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ respect $R(\vec x_i,\vec x_j)\vec ...
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33 views

Regarding a proof in Tu's 'Introduction to manifolds'

While reading Tu's differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on an open subset $U\in \mathbb{R}^n$, let $p\in U$, ...
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27 views

Question Regarding Proof of Taylor Remainder Theorem in Tu's “An Introduction to Manifolds”

The statement: Let $f$ be a $C^{\infty}$ function on an open set $U\subseteq \mathbb{R}^n$ which is star shaped with respect to a point $p=(p^1,...,p^n) \in U$. Then there are functions ...
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Show that $x_1^2+x_2^2+…+x_n^2$ defines a $C^1 (n-1 dim)$ surface in $\mathbb{R^n}$. Compute tangent space at every point

I am not sure what the idea is behind this. There is a theorem that states if a map $F:\mathbb{R}^n\to\mathbb{R}^{n-m}$ such that $dF(x)$ has rank $n-m$ at every point on a level set then that level ...
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37 views

Helmholtz decomposition of a vector field on surface

Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and ...
3
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35 views

Show that the $n\times n$ matrices with determinant $=1$ forms a $C^1$ surface of dimension $n^2-1\in \mathbb{R^{n^2}}$

I am told that I need to find a path $c(t)$ such that $c(t)=x(t), X(0)=x \forall X s.t. det X=1$. So I can show that $d/dt(f(c(t))$ at $t=0=[d_{f(c(t))}f](c'(t))]\ne 0$ My problem is how to ...
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Sufmanifold with prescribed first and second fundamental form

Is there a $2$ - dimensional submanifold $S$ of $\mathbb{R}^3$ which can be parametrized with $x : U \subset \mathbb{R}^2 \to S$ such that : $E=G=1$, $F=0$, $e=-g=1$ and $f=0$ Where $E,F,G$ and ...
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Fundamental Group and DeRahm Cohomology from Group of Covering Transformations

Old qual problem here, test tomorrow in topology and we barely got to DeRahm Cohomology so I'm not sure how to do this. Let $G$ be the group of transformations of $\mathbb{R}^3$ generated by ...
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Any oriented surface is orientable - why can we select such an atlas?

Definitions and notations: Given a surface $S$ and a surface patch $\sigma: U \subset \Bbb R^2 \to \Bbb R^3$ of $S$, we define the standard unit normal of $\sigma$ at $p$ to be (where ...
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28 views

Show every Mobius transformation $T(z)=\frac{\alpha z+ \beta}{\bar \beta z+ \bar \alpha}$ acts as an isometry of the hyperbolic disk

Consider the unit disk $\mathbb{D}=\{z: |Z| < 1\} \subset \mathbb{C}$ equipped with the hyperbolic metric $g$ induced by $1$ form $ds=\frac{|dz|}{(1-|z|^2)}$ I am trying to show that every Mobius ...
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+50

Can you determine the length of a curve by the lengths of its projections onto planes?

If $\Gamma \subset \mathbb R^n$ is 1-rectifiable, then its Hausdorff measure is equal to its integralgeometric measure. That is, $$\mathcal H^1(\Gamma) = \int\limits_{G(1,\mathbb R^n)} \int\limits_K ...
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Is there a mathematical treatment of continuously (or smoothly) deforming surfaces, or, in general, continuously deforming manifolds?

I'm wondering if there's been any attempts to construct formal mathematical descriptions of deforming surfaces, whether evolving to some steady state configuration or reacting to some pulse, force, or ...
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Prove that this is a smooth surface

S is the surface satisfying $$f(x, y, z) = z^2 + (\sqrt{x^2+y^2}-a)^2 -r^2 =0$$ where $a,r\in\mathbb{R}$ Prove that $S$ is a smooth surface. Do we differentiate with respect to $x, y$ and $z$ ...
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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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Why the well-defined of Gauss map depends on surface is orientable?

Let $S$ is a surface. Define a mapping $g:S\rightarrow S^2\subset R^3$ of $S$ into the unit sphere $S^2$ , associating to every $p\in S$ a unit vector $N(p)\in S^2$ normal to $T_pS$. Why the ...
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Ricci flow on compact surfaces flows the metric conformally

The (normalized) Ricci flow on compact surfaces is given by $$\frac{\partial}{\partial t}g_{ij}=(r-R)g_{ij}\text{ ,}$$ and in the beginning of Hamilton's paper on the topic he points out that since ...
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How to reparametrize a geodesic such that $\nabla_V V = 0$

In Nakahara's section on geodesics he says that $\nabla_V V = fV$ can be reparametrized to give $\nabla_V V = 0$ if we use the reparametrize the tangent vector components $t \rightarrow t'$ such ...
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1answer
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Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
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Why is a PDE a submanifold (and not just a subset)?

I struggle a bit with understanding the idea behind the definition of a PDE on a fibred manifold. Let $\pi: E \to M$ be a smooth locally trivial fibre bundle. In Gromovs words a partial differential ...
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Show $2xx' + 2yy' + 2zz' = 0$ for curve on sphere.

This is from Manfredo P. do Carmo's Differential Geometry of Curves and Surfaces. I have just been introduced to orientations of regular surfaces, the Gauss map and the differential of the Gauss map, ...
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Connected components of the complement of a geodesic

I came across the book "Knots, Molecules, and the Universe: An Introduction to Topology" by E. Flapan which is quite nice. In the first chapters it is discussed how one can distinguish the sphere ...
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A piecewise regular simple closed curve bisects the area of the unit sphere if and only if it has total geodesic curvature 0

How can I prove that "A piecewise regular simple closed curve bisects (this curve splits the unit sphere into two pieces, the area of which are equal) the area of the unit sphere if and only if it has ...
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2answers
39 views

Two geodesics cannot form a simple region

Suppose S is an orientable surface with nonpositive Gaussian curvature. How can I prove that two geodesics that start from the same point $p\in S$ cannot meet again at another point $q\in S$ such that ...
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1answer
24 views

Definition of a smooth function between surfaces

If $S_1, S_2 \subset \Bbb R^3$ are two smooth surfaces, then what is the formal definition of a smooth map from $S_1$ to $S_2$? I am studying from Pressley's EDG, and the definition is given only in ...
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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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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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29 views

assumptions for existence of envelope of a family of curves

Given a family of curves $F(x, y, c) = 0$ for $c$ in a range of real numbers, the envelope $E$ is the curve tangent to every member of the family. I see that it is defined by the solution of $F(x, ...
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598 views

How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows. Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be ...
3
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1answer
46 views

Correspondence between vector bundles and locally free sheaves

I am trying to look at smooth manifolds in the context of locally ringed spaces. Vector bundles have a characterization in terms of sheaves of modules: if $X, \mathcal{O}_X$ is a topological (or ...
2
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2answers
44 views

Canonical notion of parallel transport

I have a "What is the right search term?" style question: Suppose $S\subset\mathbb{R}^3$ is a surface and that we are given two points $x,y\in S$. Furthermore, take $v_x\in T_x S$ to be a tangent ...
0
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1answer
19 views

Codifferential and Hodge star

Is this true, \begin{align} \notag \delta (f * \Omega )= f \delta (*\Omega)? \end{align} $\delta$ denotes codifferential, f is a function, $\Omega$ is a k-form and * is a Hodge star operator.