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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Doubt about $n$-holed Torus and Handles

I have a doubt on the construction of the $n$-holed torus as seen on Spivak's Differential Geometry book. Spivak gives a very good argument on how to construct it: take the usual torus ...
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1answer
157 views

Gradient of a functional

Given a compact manifold with a Riemannian metric $g$, we define the total scalar curvature by $$E(g)=\int_M RdV$$ Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
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196 views

About the definition of tangent space of smooth manifold

For a smooth manifold $\mathscr M$ I have seen following definition for the tangent space at a point $m\in\mathscr M$. Define it to be $(F_m/F_m^2)^*$, where $F_m$ denotes the set of germs of smooth ...
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1answer
35 views

Calculating $d\omega$ for $\omega\in\Omega^{k}M$ explicitly for $k=2$

I am trying to explicitly calculate the exterior derivative $d\omega$ for $\omega\in\Omega^{2}M$ for a differentiable oriented manifold $M$. I know that we can express a differential $k$-form ...
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1answer
124 views

Geodesics of conformal metrics in complex domains

Let $U$ be a non-empty domain in the complex plane $\mathbb C$. Question: what is the differential equation of the geodesics of the metric $$m=\varphi(x,y) (dx^2+dy^2)$$ where $\varphi$ is a ...
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63 views

naked singularity and null coordinates

I'm trying to understand the notion of a naked singularity on a more mathematical level (intuitively, it's a singularity "one can see and poke with a stick", but I'm having troubles on how to actually ...
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0answers
88 views

characterization of the differentiable functions over a regular surface

Let $S$ be a regular surface. And let $f:S\to \mathbb R$ be a differentiable function It's not hard to prove that if $ W$ is an open set of $\mathbb R^3$ such $ V\subset S\subset W$, and $f:W\mathbb ...
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3answers
298 views

How to show that an open map $f $ implies the surjectivity of $f'$ in a dense set

Let $f$ be a $C^1$ map from $U\to \mathbb{R}^m$, where $U$ is an open set in $\mathbb{R}^n$, $n\geq m$. Then we know that if $f'$ is surjective everywhere, then $f$ is open. My question is whether ...
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263 views

Justification for this manipulation in a proof of the first variation of energy formula

As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in these ...
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3answers
254 views

curvature of the boundary of a convex set is positive

Let's consider $J\subset \mathbb R^2$ such that J is convex and such that it's boundary it's a curve $\gamma$. Let's suppose that $\gamma$ is anti-clockwise oriented, let's consider it signed ...
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1answer
75 views

Gauss curvature of graphs of a function

Suppose that $f,g:\mathbb{R}^{2}\rightarrow \mathbb{R}$ are smooth functions, with $g(u,v)\geq f(u,v)\geq 0$ for all $u, v \in\mathbb{R}$ and with f(0,0)=g(0,0)=0. Let $κ_{f},κ_{g}$ be respectively ...
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1answer
58 views

how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$?

I'm digging my head to prove the following relation: ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$ where $\theta$ is the Maurer-Cartan form and ...
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2answers
88 views

Tangent vectors on SO(2)

Parametrization of group SO(2) can be given as the following; $$ \begin{pmatrix} \cos( \theta) & \sin(\theta)\\ - \sin(\theta) & \cos(\theta) \\ \end{pmatrix} ...
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1answer
140 views

Heuristics for the Yamabe Problem

I am trying to understand the Yamabe problem, and I was naturally lead to a question: given a manifold with a Riemannian metric on it, why is it interesting to find a conformally related metric that ...
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4answers
79 views

Why $GL(n+1,\mathbb{C})$ is compact?

I'm trying to prove that: The set of all lines in $\mathbb{C}^{n+1}$ ($\mathbb{C}\mathbb{P}(n)$) is a complex manifold. I'm knowing that: If a compact group $G$ acts on $X$ transitively and ...
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1answer
62 views

Convex Curve Parametrization

How can I parametrize a convex plane curve using the angle $\theta$ between the tangent line and the $x$-axis?
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94 views

Lie bracket in local coordinates. Find the formula $c^{k}$ in terms of $a^{i}$ and $b^{j}$

This is from T.U Loring's manifold book. I tried. But I didnt do the question. Please show me how to solve instructively and explicitly. I want to learn this topic. Thank you for help.
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1answer
86 views

Restriction map on a compact orientable manifold without a boundary.

I have the following problem: Let $M$ be and $n$-dimensional compact oriented manifold without boundary. Let $p\in M$ be and point and let $M_p=M\backslash\{p\}$. Let $j:S^{n-1}\to M_p$ be the ...
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2answers
185 views

Topological space M with partition of unity--->M paracompact. John Lee Problems

Suppose $M$ is a topological space with the property that for every open cover $X$ of $M$, there exists a partition of unity subordinate to $X$. Show that $M$ is paracompact.
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1answer
93 views

How do I show this is strictly positive without using matrices.

Suppose $ds^2 = Edu^2 + 2Fdudv + Gdv^2$ is the first fundamental form of some regular surface patch, show $EG - F^2 > 0$ for each point on the surface patch. So technically I could put $E$, ...
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53 views

Cohomologies of $\mathbb R^n$ with rational differential forms

We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
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1answer
340 views

principal curvature and its relationship to second fundamental form

In the Wikipedia definition of principal curvature, it says that the principal curvatures are the eigenvalues of the symmetric matrix $\left[I\!I_{ij}\right] = \begin{bmatrix} ...
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1answer
484 views

Tangent space at the identity element of a lie group

Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth . Now by identifying ...
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205 views

Definition of lie bracket of vector fields

The Jacobi-Lie bracket or simply Lie bracket, $[X,Y]$, of two vector fields $X$ and $Y$ is the vector field such that $[X,Y](f) = X(Y(f))-Y(X(f)) \,.$ ...
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183 views

How we do actually compute the topological index in Atiyah-Singer?

I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing ...
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2answers
216 views

Generally covariant Klein-Gordon equation

Consider a 4-dimensional smooth manifold $M$ on which there is a Lorentzian metric $g_{ab}$ and a function $\phi$ satisfying the following two equations (in abstract index notation): \begin{equation} ...
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1answer
147 views

Lie bracket of vector fields and differential of diffeomorphism in its definition

To make the connection to the Lie derivative, let $t \mapsto \Phi^V_t$ be the 1-parameter group of diffeomorphisms (or flow) generated by the vector field $ V $. The differential $ d\Phi^V_t $ ...
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102 views

Principal connection & curvature

Let $(P, \pi, B)$ be a principal $G$-bundle over $B$ and $\omega$ a principal connection. Then the curvature is defined as $$ \Omega_\omega = d \omega + \frac{1}{2} \omega \wedge \omega$$ With the d ...
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1answer
74 views

The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $ ∂/∂u|p,∂/∂v|p $ into $TpR_{3}. $ Find $(α_{i}, β_{i}, γ_{i})$

Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?
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53 views

New operation on the functions of a Riemannian manifold

Let $(M,\rho)$ be a Riemannian manifold and let ${\cal F}(M)$ denote the algebra of smooth functions of $M$. Given $f\in{\cal F}(M)$, denote by $X^f$ its gradient vector field with respect to $\rho$. ...
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1answer
50 views

Show that anecessary and sufficient condition for $x_{p}$ to be tangent to $S^{n}$ at $p$

Please help me! How do I solve this problem? I didnt produce any idea because I didnt understand this topic properly. Thus, please can you explain the solution explicitly? Thank you for help:)
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1answer
86 views

Projecting external points to a circle: Distance order preserving?

Given a circle, and a set of points $A$ that lie external to the circle; I perform the following simple operation: I compute the point of intersection of the i) circle and the ii) line joining each ...
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84 views

Geodesic on a Hilbert manifold

Given a Hilbert manifold $\mathcal H$ (always using the natural Hilbert inner product) and a geodesic $\Gamma(t)$ in this manifold, can one show that the projection of this geodesic onto a submanifold ...
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1answer
58 views

Differentiation of product of one parameter subgroups

Let $G$ be a Lie group. Let $\gamma,\rho : I \to G$ be two smooth curves such that $\gamma(0) = \rho(0) = I$ the identity and $\gamma'(0) =A$ and $\gamma'(0) = B$ for some $A,B \in G$. My ...
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votes
1answer
90 views

How to see the triangulation of an object

I am reading simplicial complex in Algebraic topology by Spanier. I read the definition of triangulation of a space and polyhedral. I cant get a picture about what we mean by triangulating a space and ...
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1answer
117 views

How does the holonomy act on the tangent space at a point?

Suppose $(X,h)$ is a compact $n$-dimensional Hermitian manifold, with holonomy group $H$. Now we know,since $X$ is a complex manifold, that $H\subset U(n)$, and that there is a representation of $H$ ...
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1answer
570 views

Quotient theorem for tensors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
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2answers
200 views

Projective Plane and Projective Space

I have already heard of a $n$ dimensional manifold called the projective space which is the set of all lines through the origin of $\mathbb{R}^{n+1}$. Spivak presents in his Differential Geometry book ...
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votes
1answer
60 views

Example of certain curve

It's known that a smooth curve of speed one (i.e parametrized by arc length) in R^3 with non-zero curvature and zero torsion (everywhere) is contained in a plane. I need to prove (by giving an ...
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1answer
55 views

injectivity radius of hyperbolic manifolds

It is known that one can construct hyperbolic surfaces that have arbitrarily large injectivity radii. Is this true for higher dimensional hyperbolic manifolds? In particular I'm interested in the case ...
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2answers
506 views

Parallel Transport along a curve

We had this homework assignment for our geometry course, and we couldn't figure it out, any ideas on how to do this: Consider the Poincare model of Lobachevsky plane, $H^2=\left\lbrace{ ...
2
votes
2answers
174 views

Finding the degree of a map

I am having trouble computing the degree of a certain map using the fact that $f: N \rightarrow M$ where $M$ and $N$ are both $n$-dimensional manifolds induces a homomorphism between the nth ...
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0answers
191 views

Metric on Steifel and Grassmannian manifolds generalizing Fubini-Study

If $F$ is $\mathbb{R}, \mathbb{C}$, or $ \mathbb{H}$, the Grassmannian manifold $G_k(\textbf F^n)$ is the space of all $k$ dimensional subspaces of the $n$ dimensional vector space $F^n$. The Stiefel ...
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1answer
57 views

Confusion regarding uniqueness of Levi-Civita connection

Assuming a Levi-Civita connection exists it is uniquely determined. Using $\nabla g = 0$ and the symmetry of the metric tensor $g$ we find: $ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(Y,X)) = ...
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461 views

Tangent bundle of a quotient by a proper action

Given a compact group $G$ acting freely on a manifold $X$, is there a "nice" way to describe the tangent bundle of the quotient $X/G$ (when it is a manifold)? In the case the group $G$ is finite, or ...
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377 views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
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299 views

Euler Characteristic of surface with boundary puncture.

I am studying a course on differential geometry. I saw the formula for the euler characteristic of a surface with $g$ holes and $b$ boundaries components and $n$ punctures is $2-2g -b +n$. In ...
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2answers
400 views

Determine the result of parallel translating the vector $(0, 0, 1)$ once around the circle $x^2+ y^2=a^2, z=0$

can any one solve this problem Q) Determine the result of parallel translating the vector $(0, 0, 1)$ once around the circle $x^2+ y^2=a^2, z=0$ .On the right circular cylinder $x^2 + y^2 ...
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72 views

Diffeomorphisms between factors in diffeomorphic product manifolds

Let $M$, $N$ and $P$ be three smooth manifolds such that $M \times N$ is diffeomorphic to $M \times P$. I need to know about some conditions under which one can deduce that $N$ is diffeomorphic to ...
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203 views

Riemannian curvature and its application on covariant derivative of tensors

This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...