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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Integration of a differential form

Let $\omega$ be a $2$-form on $\mathbb{R}^3-\{(1,0,0),(-1,0,0)\}$, $$\omega=((x-1)^2+y^2+z^2)^{-3/2}((x-1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)+ ((x+1)^2+y^2+z^2)^{-3/2}((x+1)dy\wedge dz+ydz\wedge ...
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1answer
279 views

Kernel of adjoint of Lie algebra

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. The adjoint representation of the Lie algebra $\mathfrak{g}$ is defined as: $$ \text{ad: } \mathfrak{g} \rightarrow ...
4
votes
1answer
673 views

Regular value: intuition about surjectivity condition

Let $f:M\rightarrow N$ be a smooth function between two smooth manifolds. A $\textit{regular point}$ is a point $p\in M$ for which the differential $df_p$ is surjective. What does the surjectivity ...
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votes
2answers
254 views

Zero sections of any smooth vector bundle is smooth?

Could any one give me hint how to show that the zero section of any smooth vector bundle is smooth? Zero section is a map $\xi:M\rightarrow E$ defined by $$\xi(p)=0\qquad\forall p\in M.$$
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votes
1answer
196 views

Lie algebra of a Lie subgroup

Let $G$ be a Lie group and $H$ a Lie subgroup of $G$, i.e. a subgroup in the group theoretic sense and an immersive submanifold. Let $\mathfrak{g}$ and $\mathfrak{h}$ be the associated Lie algebras. ...
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1answer
98 views

functions with positive Laplacian

Information about the class of compactly supported smooth functions $u$ on $Ω\subset R^n$ such that $Δu≥0?$ Do a significant class of such function exist? NB. This type of function may be useful in ...
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0answers
71 views

Pullback of conformal killing field via conformal map

This is my first time posting on this forum, so to start with, it's good to meet you all and thanks in advance for the help! My question is as follows. Suppose I have two semi-riemannian manifolds of ...
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1answer
481 views

left-invariant vector field: counterexample

Let $G$ be a Lie group, $L_g$ the left-translation on this group with differential $d L_g$. A vector field $X$ on $G$ is called left-invariant if $$ X \circ L_g = d L_g \circ X \quad \forall g \in ...
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0answers
107 views

Given a vector field all of whose integral curves are closed, is the period a smooth function?

I am reading about the energy-period relation for Hamiltonian Systems. In Weinstein's formulation (cf. Abraham, Marsden, Foundations of Mechanics 2nd Ed, page 198) this relation amounts to: ...
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2answers
698 views

How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text ...
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vote
1answer
179 views

Extension with harmonic function

Let $\Omega$ be a domain of $R^n,$ let $\omega$ be open subset of $\Omega$ and let $\theta \in W^{2,\infty}(\omega).$ I am wondering about the existence of a function $\tilde{\theta} \in ...
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53 views

Calculation of |[X,Y]^V|

I want to follow the proof of Theorem 3.1 in "On Eschenburg's Habilitation on Biquotients" - Wolfgang Ziller. The situation is as follows: $Q$ is a biinvariant metric on $G$. So from the ...
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votes
1answer
249 views

Continuity equation on manifolds

Mass conservation is usually written as $$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho \boldsymbol v) = 0$$ $\rho$ is the density and $\boldsymbol v$ is the fluid velocity. My attempt ...
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2answers
387 views

What exactly is a manifold?

Wikipedia's "Simple English" entry describes a 2D map of the Earth as a manifold of the planet Earth. Does this mean that in mathematics a manifold is essentially a representation of something that ...
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votes
1answer
181 views

Relation between Lie algebras and Lie groups

I am a little confused as to how to compute generally the Lie algebra of a Lie group and viceversa, namely the Lie groups (up to diffeomorphism) having a certain Lie algebra. The way I did this for ...
5
votes
0answers
67 views

Inner product of $p$-forms [duplicate]

Possible Duplicate: Extension of Riemannian Metric to Higher Forms I have no problems with understanding the inner product of 1-forms on a Riemannian manifold. We have a metric tensor, it's ...
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1answer
723 views

Riemannian Volume Form

There is a following exercise in my text: Let $S^n$ be $n-$ dim sphere in $R^{n+1}$ with inclusion function $i:S^{n}\to R^{n+1}$. Let $$\omega=\sum_{i=1}^{n+1}(-1)^{i-1} x_i dx_1 \wedge... ...
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1answer
95 views

Tangent field of a strictly convex closed curve

Let $\gamma:[0,a] \to \mathbb{R^2}$ be a simple smooth closed curve with curvature $\kappa (t) \neq 0$ $\forall t \in [0,a]$. Prove for each $\vec{u} \in S^1$ there exists a unique $t_0 \in [0,a]$ ...
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votes
2answers
288 views

The orientation of quotient manifold

If $T$ is a torus and $\mathbb Z_2$ acts on it by $(z_1,z_2)\rightarrow(z^{-1}_1,-z_2)$, then is $T/\mathbb Z_2$ orientable?
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votes
2answers
357 views

Question about Angle-Preserving Operators

This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given ...
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votes
2answers
215 views

Riemannian volume form on surface of a smooth function

It should be easy calculation exercise in my text, but I am afraid I am a little bit stuck on the concept of the question. Let $f:\Bbb R^n\to \Bbb R$ be a smooth function. Consider graph $X$ of $f$ ...
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0answers
385 views

Torsion in two dimensions?

This question is about the notion of a connection with torsion in differential geometry, i.e., a connection that is not Levi-Civita. (It's not about the torsion of a curve in three dimensions.) ...
2
votes
1answer
102 views

Example of a proper homotopy between smooth functions on manifolds

Let $h:S^{n-1}\to S^{n-1}$ be $C^{\infty}$ map. How to prove that a function $F: S^{n-1}\times[0,1]\to S^{n-1}$ given by $$F(v,t)=(\cos{\pi t})v+(\sin{\pi t})h(v)$$ is proper $C^{\infty}$ map?
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1answer
582 views

Continuous maps between compact manifolds are homotopic to smooth ones

If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$. Seems to be fairly basic, ...
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1answer
193 views

variation problem of constrained area and minimized distance

$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$ The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ...
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1answer
155 views

How to calculate first variation of functionals defined on curve

I wonder if someone can explain to me how one goes about to calculating the first variation of functions defined on curves. For example, if $C$ is a curve in $\mathbb{R}^3$, and the functional $$F(C) ...
2
votes
1answer
149 views

Integrals of forms are equal implies they differ by $d\mu$

The problem is about the proof of the following result. If $\omega_1,\omega_2 \in \Omega^n_c(X)$ (where $X$ is smooth manifold) are such that $\int_X\omega_1=\int_X \omega_2$ then there is ...
2
votes
1answer
544 views

Proof of Gauss's Lemma (Riemannian Geometry version)

I was self-learning Do Carmo's Riemannian Geometry, there is a step in the proof of Gauss's Lemma what I can't quite figure out. Since $d\,\exp_p$ is linear and, by the definition of $\exp_p$, ...
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votes
2answers
1k views

Proving that the general linear group is a differentiable manifold

We know that the the general linear group is defined as the set $\{A\in M_n(R): \det A \neq 0\}$. I have a homework on how to prove that it is a smooth manifold. So far my only idea is that we can ...
6
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1answer
745 views

Structures on torus

Quotienting $\mathbb R^2$ by different lattices isomorphic to $\mathbb Z^2$, we get different tori. Somehow I think of the tori as having different "structures", but thinking more about it, I am not ...
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1answer
131 views

There is a closed non-zero $n$-form on $\text{GL}(n, \Bbb{R})$

How to Prove that There is a closed non-zero $n$-form $\omega$ on $\text{GL}(n, \mathbb{R})$ which is left and right invariant.
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1answer
99 views

Algebraic topology involved in the 1/4-pinched sphere theorem?

Can anyone familiar with this theorem and its proof let me know how much algebraic topology is involved, and where specifically? I am familiar with a lot of differential geometry, but not many of the ...
2
votes
2answers
873 views

What do level curves signify?

Suppose I have a function $z=f(x,y)$, say like $z=\sqrt{x^2+y^2}$. By fixing some value for $z$ and varying all possible $x$ and $y$, we would get a level curve of $z=f(x,y)$. By changing values for ...
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3answers
553 views

Boundary of product manifolds such as $S^2 \times \mathbb R$

Simple question but I am confused. What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$? What would be the general way to evaluate the boundary of a product manifold? Thanks for the ...
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0answers
92 views

Analog of a tubular neighborhood for an embedded wedge sum

If you have some embedding of a path connected topological space wedge of spheres $N$ into a compact simply connected smooth $n$ manifold $M$ (like a sphere for example), then is there some kind of ...
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1answer
91 views

Differential Geometry for C^n

Does anyone know a good resource to read up on differential geometry for 2 complex dimensions with an anti-symmetric metric tensor?
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2answers
245 views

How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$. Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$. Prove: Suppose $f\in C^\alpha(M)$ satisfies ...
4
votes
2answers
358 views

Moving to a conformal metric

Given a generic 2-dimensional metric $$ ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2 $$ what is the change of coordinates that move it into the conformal form $$ ...
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vote
2answers
432 views

Coordinate-free method to determine local maxima/minima?

If there is a function $f : M \to \mathbb R$ then the critical point is given as a point where $$d f = 0$$ $df$ being 1-form (btw am I right here?). Is there a coordinate independent formulation of ...
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0answers
211 views

Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
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vote
1answer
123 views

Degree of Hessian surface invariant under linear transformations?

Given a surface $V(f) \subset \mathbb{P}^n$ for a homogeneous polynomial $f$ of degree $d$ on $\mathbb{P}^n$ and a linear transformation $g \in SL(n+1)$. Is the degree of the Hessian $H_f = V(\det ...
7
votes
1answer
1k views

General form of Integration by Parts

This is a question just out of interest to know the power of integration by parts. There are various level of integration by parts. What are some of the most general form of integration by parts? I ...
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votes
1answer
79 views

What are the allowed dimensions for vector fields?

When one first encounters the concept of vector field, especially in physics, it is often presented just as n-tuple of numbers $(x_1, x_2, \ldots , x_n)$ prescribed to each point. In this manner $n$ ...
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2answers
773 views

Converse to Inverse Function Theorem?

A fairly general form of the Inverse Function Theorem is: Suppose $X, Y$ are Banach spaces, $U \subset X$ is open and $f:U \to Y$ is continuously differentiable. If for some $x \in U$ the derivative ...
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0answers
324 views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
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2answers
743 views

Learning differential/Riemannian geometry for PDEs

I know there have been threads on which books to learn DG/RG from but hopefully this is sufficiently different to avoid closure. Can anyone recommend a book to learn DG/RG (whichever is appropriate) ...
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1answer
326 views

interior product - proof of the basic fomula

How would you prove the interior product formula? Namely for $\omega \in \Omega^k (X), \mu\in \Omega^l(X)$, where $X$ is smooth manifold with vector field $v$ we have $$i(v)(\omega ...
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1answer
84 views

How do I find the number of vertices on a planar diagram of a surface?

Cheers, I have a question which I just do not seem to see the answer for: I am proving the classification theorem for compact surfaces and use planar diagrams as representation of the surfaces. I ...
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1answer
161 views

The number of geodesics of a complete Riemann manifold with non-positive sectional curvature

There is a theorem of Cartan which states that if $M$ is a simply connected, complete Riemann manifold, and that the sectional curvature is everywhere $\leq 0$, then any two points of M are joined by ...
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0answers
61 views

Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one ...