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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Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a ...
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8 views

Can the nabla symbol be used for a convariant derivative?

Can $$ \nabla_{\nu} A^{ \mu\nu} $$ represent a covariant derivative with respect to $\nu$? If not what can it be? I'm reading a textbook on General Relativity, and such operations appear without any ...
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37 views

Parametrization of the sphere and the torus.

Is there a way to find easily the parametrization of the sphere and the tore ? I see on wikipedia that for the sphere it's $(x,y,z)=(\sin \theta\cos \varphi,\sin\theta\sin\varphi,\cos\varphi)$ with ...
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10 views

Gauss curvature and bisectional curvature

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where X is a Kahler manifold. Then is it correct that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega)$$ where ...
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35 views

What am I doing wrong when calculating this pullback?

Let $\omega = \sum_{j=1}^{n+1} x_j dy_j - y_j dx_j $ be a differential form on the sphere $S^{2n +1}$. Let $G = Z_2$ be the group acting on the sphere. I want to apply the following proposition to ...
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29 views

Orientability of differantiable manifold of orthogonal matrices

I want to find out if differentiable manifold of matrices $M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9$ is orientable. It is only worth proving that orthogonal ...
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38 views

Why say “exists smooth structure” instead of “is a smooth manifold”?

As some of you know I started to learn differential geometry some weeks ago. Now I came across this theorem (you can find it in Lee's Intro to Smooth Manifolds but it is in many other books also: ...
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20 views

When is a stable domain in a minimal surface area minimizing?

A stable domain $D$ in a minimal surface $S\subset \mathbb{R}^3$ is a domain for which the area-functional $A(t):=\int_{S_t}dS_t$ has non-negative second derivative, i.e. $A''(0)\geq 0$, for all ...
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1answer
32 views

Exterior derivative of complex differential form

I have this question, from several complex variables: Start with the differential form: $$\omega(z)=\sum_{\nu=1}^{n} \frac{(-1)^{\nu-1}\bar{z}_{\nu}}{|z|^{2n}} d\bar{z}[\nu] \wedge dz, $$ where ...
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33 views

Demonstration of a basic formula involving differential forms

I'm writing some notes on Lie Groups and I'm not sure if I should demonstrate this formula or not. Assume $\omega$ is a differencial form and $X,Y$ fields con a Manifold M, is there a simple way to ...
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Whether there are any differential structure in the homology or homotopy of real coefficient ? [on hold]

I listen a report , the reporter says that the group of all closed curves of a Riemannian manifold has differential structure . So I guess the homology or homotopy of some suitable coefficient have ...
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Finding braches of equilibria

Consider the system of two equations in three variables $(x,y,z)$: $$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$ The point $x^* = ...
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2answers
48 views

Orientability of surfaces

How to prove that a surface is orientable? Is it true that the union of two orientable surfaces is orientable? How to prove that? For example, is the union of the hemisphere $$z = \sqrt{1 - x^2 - ...
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101 views

Orientability of $m\times n$ matrices with rank $r$

I know that $$M_{m,n,r} = \{ A \in {\rm Mat}(m \times n,\Bbb R) \mid {\rm rank}(A)= r\}$$is a submanifold of $\Bbb R^{mn}$ of codimension $(m-r)(n-r)$. For example, we have that $M_{2, 3, 1}$ is ...
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1answer
40 views

Solving the Euler-Lagrange equations for geodesics

I am trying to find geodesics on the following metric: $ds^2 = dx^2 + x^2 dy^2$ Setting $dx \rightarrow \dot{x}, dy \rightarrow \dot{y}$ in $ds^2$ i get following Lagrangian: $L = \dot{x}^2 + x^2 ...
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21 views

Christoffel Symbols - Showing Equality

Using: $$\sum_{j} g_{ij}g^{jk}=\delta_i^k$$ $$\Gamma_{ijk}+\Gamma_{jik}=\frac{\delta g_{ij}}{\delta u^k}$$ $$\Gamma_i{^j}_k = \Gamma_k{^j}_i = \sum_{l} g^{jl}\Gamma_{ilk}$$ Show that: ...
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50 views

If $\gamma :[a,b]\rightarrow \mathbb{R}^3$ is smooth then $\gamma(t)=x$ has finite number of solutions

Let $\gamma :[a,b]\rightarrow \mathbb{R}^3$ be a smooth curve ($\gamma$ is differentiable with $\gamma'(t)\neq \mathbf{0}$ for all $t\in[a,b]$). Show that, for $x\in\mathbb{R}^3$, the equation ...
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1answer
37 views

Frobenius theorem

I came across the following conclusion in a textbook, but can't really understand it. I would be grateful if anyone could elaborate: Assume that we have two linearly independent vector fields ...
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17 views

open set in tangentspace induces open set in tangentbundle (for homogeneous spaces)

Let $M=G/K$ be a homogeneous space with a $G$-invariant riemannian metric $<.,.>$. Then $G$ defines an action on $TM$ by derivatives. Let $p=eK \in M$. Assuming I have a set $V_p \subset T_pM$ ...
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22 views

Orientation form on manifold cut out by $m$ functions

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function. If $a$ is such that $f$ has surjective derivative at all points in $f^{-1}(a)$ then this is an $n-m$ dimensional manifold $X$. I'm trying ...
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1answer
21 views

Functions with nonnegative laplacian on Rimannian manifold.

I am doing the exercises in Do Carmo's "Riemannian Geometry". I am stuck on exercise 3.12 which states the following: Let $M$ be a compact orientalbe Riemannian manifold which is also connected. Let ...
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1answer
39 views

Vector Fields on $\mathbf R^2$ [on hold]

Let $X : \mathbf R^2 \to \mathbf R^2$ be a no-zero smooth vector field. I want to show (without background about vector bundles or manifolds, just if possible differentiable calculus in $\mathbf R^n$) ...
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Why does the coordinate transformation from Cartesian coordinates leads to an additional term in the biharmonic operator in spherical coordinates

I am trying to solve a problem in physics where the biharmonic operator is involved. I think that the bihahmonic operator can be obtained by taking twice the Laplace operator, such that $\nabla^4 f = ...
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0answers
21 views

Isothermal coordinates [duplicate]

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function? I know that this metric is a special kind ...
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2answers
39 views

Calculating the differential of the quotient map using curves

We can view the projective space $P(\mathbb R^n)$ as the quotient of $S^n/\sim$ where $x \sim y$ if and only if $x = -y$. The quotient map $F: S^n \to P(\mathbb R^n)$ is the map $x \mapsto [x]$ ...
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1answer
19 views

Is every hyperbolic isometry the restriction of an orthochronous Lorentz transformation?

I know that every isometry of the sphere $\Bbb S^2$ is the restriction of some $A \in {\rm O}(3,\Bbb R)$: namely, if $A_0:\Bbb S^2\to \Bbb S^2$ is an isometry, then $A_0 = A\big|_{\Bbb S^2}$ where ...
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41 views

Killing fields on product metrics

Let $(M_i,g_i)$ be Riemannian manifolds, $i=1,2$. (Save Euclidiean factors) Is it true that a Killing field $Z$ on $(M_1\times M_2,g_1\times g_2)$ will split as a sum of Killing fields $Z=X+Y$, where ...
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Self-commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set ...
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59 views

Elementary properties of gradient systems

Consider $x_0\in\mathbb{R}^n$ and a $C^{1,1}$ function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ (that is, a differentiable function whose gradient is Lipschitz function). Consider the system $$ ...
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The lenght of rectifiable curve in $\mathbb{R}^n \setminus B[0,r]$ that connects antipodes points.

This question is from my homework, here it goes: Let $\gamma \colon [a,b] \to \mathbb{R}^n \setminus B[0,r]$ be a rectifiable curve such that $\gamma(a)=-\gamma(b)$. Using euclidean norm prove that ...
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2answers
51 views

Laplacian of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how I can write $\Delta^M f$ in terms of $\Delta^{S}f$ ? ((i.e the relation between ...
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0answers
18 views

Gradient and Laplacian of a submanifold

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$. Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. Similarly ...
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1answer
52 views

Given a $1$-form $\omega$ on $\Bbb R^n$, is there a connection whose torsion is $T(X,Y)=\omega(X)Y-\omega(Y)X$?

Consider $(R^n, g_0 )$, where $g_0$ is the Euclidean metric, and a differential $1$-form $\omega$ on $R^n$. Can this differential form define a connection on $M=R^n$ such that its torsion is ...
2
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1answer
50 views

Understanding algebraic operations on vector bundles

Let $X$ be a smooth manifold with vector bundles $V$ and $W$. I'm trying to understand how we can construct in general a vector bundle $V \otimes W$. In particular what manifold structure do we give ...
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19 views

Lie bracket and local group

How to prove this identity? X and Y and smooth vector field on smooth manifold M; $\theta_t$ is the local group (one-parameter group of diffeomorphism) of Y. ...
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0answers
30 views

Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
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1answer
21 views

Is an isometry between compact boundaryless embedded surfaces necessarily a rigid motion of $\mathbb{R^3}$?

A friend and I were discussing this and related questions as part of pre-exam revision, and we don't know how to answer this particular question (could not think of a proof or counterexample). Any ...
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1answer
25 views

If F is a diffeomorphism and $F_*$ preserves dot products, then F is an isometry.

Exercise from O'neill's book ELEMENTARY DIFFERENTIAL GEOMETRY (p.121) $If \quad F:R^3\to R^3\quad is \quad a\quad diffeomorphism\quad such \quad that \quad (its\quad tangent\quad map)\quad F_*\quad ...
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13 views

Area form of $M^2 \subseteq \Bbb R^4$: does $(-1)^{i+j-1}\det_{i'j'}(n,\nu) = dx^i \wedge dx^j$?

This is a follow up question from this one. I'm asking separately since one way or another, that one is sort of answered. Now, we know that if $M^{n-1} \subseteq \Bbb R^n$, then the volume element ...
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References on the moduli space of flat connections as a symplectic reduction

In their Yang Mills equations over Riemann surfaces paper, Atiyah & Bott famously remark that the moduli space of flat connections on a principal bundle over a compact orientable surface may be ...
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1answer
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Specific values of paramaters for which curve is closed

In my study of curves, I encountered this family of parametrized curves in $ \mathbb{R}^2 $ $ \cosh(y)=-A\cos(x)+B $ for real parameters A and B such that $ 0 < |A| < 1 $ My problem is to ...
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Determining whether a Lie group contains more than one conjugacy class of subgroups of a particular isomorphism type

Suppose I have a Lie group $G$. How can one determine whether there is more than one conjugacy class in $G$ of subgroups isomorphic to a given Lie subgroup $H$? Put another way: Fix a Lie ...
4
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2answers
59 views

Dimensions of immersions vs embeddings

Let's say that you have a manifold which you know can be immersed in $\mathbb{R}^n$. Is there a $k$ such that you can say, for sure, that the manifold is embedded in $\mathbb{R}^{n+k}$? I imagine that ...
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1answer
42 views

Why the topological dimension of C is 2?

From what I know, the topological dimension of a set has to do with open sets covering it, homeomorphic to R^{n}. Then we can cover C with balls, for instance,of ...
2
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0answers
34 views

Infinite surface area

I am reading an article (reference: http://www.jstor.org/stable/1971139?seq=1#page_scan_tab_contents), and in the proof of the main theorem, the author states that "it is a fact that complete, ...
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1answer
80 views

Area form for $M^2 \subseteq \Bbb R^4$

We know that in general, given a orientable hypersurface $M^{n-1} \subseteq \Bbb R^n$, the volume form on $M$ is given by $$dM = \sum_{i=1}^n(-1)^{i-1}n_i\,dx^1 \wedge\cdots\wedge \widehat{dx^i}\wedge ...
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1answer
24 views

Construct vector field along a curve

Let $M$ be a smooth manifold. I am trying to construct a (piecewise smooth) vector field $V$ along a curve $c$ that takes on prescribed values $K_{i}$ at times $t_{i}$. Say we construct V along ...
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2answers
48 views

Gradient vector proof

Question: Prove that a normal vector to the surface $f(x,y) = \sqrt {xy}$ at any point on the surface is perpendicular to the line joining the point to the origin. I am not sure how to do this. ...
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1answer
16 views

How to find the normal line of $z = 4x^2 + y^2 - 78$ at $(2,1,-61)$?

How do I find the normal line of $z = 4x^2 + y^2 - 78$ at $(2,1,-61)$? I have found that the tangent plane is $z-16x-2y=95$ but I don't know how to find the normal line. The answer is: $$\frac{2 ...
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13 views

Compact-Open Topology for Space of C^{r} -sections

Given a smooth fibre bundle $\pi: X \rightarrow M$. What is the definition of compact open $C^{r}$-topology on the space of $\mathcal{C}^{r}$-sections?