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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Submanifold of $\mathbb R^n$ : projections onto tangent spaces

Let $M$ a submanifold of $\mathbb R^n$, for all $x$ in $M$, let $\pi_x:\mathbb R^n\rightarrow T_xM$ the orthogonal projection onto the tangent space $T_xM$ of $M$ at $x$. How could you show that for ...
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220 views

Definition of Sectional Curvature

do Carmo gives a definition of sectional curvature as follows: $$K(x,y) = \frac{\langle R(x,y)x,y\rangle}{|x\times y|^2}$$ where $x,y \in T_pM$ are linearly independent vectors. My question: The ...
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251 views

Definition of Willmore energy

The MAA has posted to its facebook page a link to an article about a recent proposed proof of what is called the Willmore conjecture, after Thomas Willmore. Wikipedia's article titled Wilmore ...
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259 views

Frobenius Theorem; Slices

Good Night. I am studying the Frobenius theorem. I'm reading the book Foundations of differentiable manifolds and Lie Groups; Frank Warner. In the first third part of the statement is written, "is a ...
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2k views

implicit equation for “double torus” (genus 2 orientable surface)

The embedded torus in $\mathbb R^3$ can be described by the set of points in $(x,y,z)\in \mathbb R^3$ satisfying $T(x,y,z)=0$, where $T$ is the polynomial ...
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195 views

differentiating under the integral sign

Suppose I have a moving curve $\alpha:I\times[0,T] \to \mathbb{R}^n.$ Its length is $$\int_\alpha ds = \int_I |\alpha_x|dx.$$ If I want to find the time derivative of this, I guess I differentiate ...
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84 views

A few basic questions on curves in $\mathbb{R}^n$

I have a number of basic questions. 1) Consider a curve $\gamma:I \to \mathbb{R}^n$. I understand that if the arclength is $s$, the change of variables requires $ds = |\gamma_x|dx$. But what does ...
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179 views

derivative of path on a sphere

I'm new here so sorry if this is a really silly question but I can't solve it myself. If I have two points on a unit sphere, A and B, and the shortest path from A to B over the surface of the sphere, ...
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275 views

De Rham cohomology of the euclidean space without n lines

How can I compute the de Rham cohomology of $\mathbb{R}^3$ minus n lines through the origin? I would like to do it with the Mayer-Vietoris sequence (which is the only thing I know to calculate ...
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276 views

How to show that “spheres” are diffeomorphic?

Can anyone provide a diffeomorphism between these "spheres": $\mathbb{S}^2$ and $\{(x,y,z)\in \mathbb{R}^3: x^4+y^2+z^2=1\}$? PS: If you know a result that can solve this problem, I would be glad to ...
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105 views

differential and arc length notation question

Suppose $\alpha$ is a time dependent curve so that $\alpha:[0,T]\times I \to \mathbb{R} ^n$. I am a bit confused as to what the meaning of the expression $\partial_t(ds)$ is, where $ds = |\partial_x ...
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298 views

Relationship between tangent and normal vector

Why is this true? If $\alpha$ is a time dependent curve, $T$ is the unit tangent and $N$ is a normal field along $\alpha$, then $$\langle \partial_s N, T \rangle = -\langle N, \partial_s T \rangle$$ ...
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204 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...
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1answer
97 views

Linear independence regarding Exterior Power .

I have been trying to learn the proof of dimension of exterior power from this text : http://www.thehcmr.org/issue1_2/poincare_lemma.pdf.( Page 16) I am not able to understand the part of linear ...
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953 views

What does it mean to say a boundary is $C^k$?

I need a explanation on what does it mean to say a boundary is $C^k$. Can anyone help me please. And also need some explanation on how to straighten boundary ?
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369 views

differential form

one form $\alpha$ over a smooth manifold is non vanishing means for every $p\in M$, $\alpha_p\neq 0$. But $\alpha_p$ is linear map $T_M\to \mathbb R$, hence $\alpha_p(0)=0$. So confusion arises ...
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360 views

notation of differentiation in differential geometry

I can't wrap my head around notation in differential geometry especially the abundant versions of differentiation. Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I ...
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1k views

Proof that angle-preserving map is conformal

Let $\phi: S \to \bar{S}$ be a diffeomorphism between two surfaces in $\mathbb{R^3}$. Such a map is called conformal if for all $p \in S$, and $v_1, v_2 \in T_p(S)$ (the tangent plane) we have ...
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1answer
210 views

Tensored vectorspaces isomorphic to the endomorphisms [duplicate]

Possible Duplicate: Understanding isomorphic equivalences of tensor product I have the following question: Let $V$ be a vectorspace with an inner product $<.,.>$. Let $V^{*}$ be its ...
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417 views

Explicit computation of the Hodge codifferential

Question I'm given a Laplacian $\Delta_n=-4y^2 \cdot \frac{\partial^2}{\partial\bar{z} \partial z} + 4 iny \cdot \frac{\partial}{\partial\bar{z}}$, and I want it to be the Laplace operator associated ...
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1answer
59 views

Behavior of $L^2$-spaces under conformal variation

Consider a Riemannian manifold $(M,g_0)$ which is the interior of a compact manifold $(\overline{M}, \overline{g})$. I'm interested in a kind of conformal variation of the background metric $g_0$. ...
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214 views

positive non-constant harmonic function $f $ in $L^1(M)$ on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, does there exists a positive non-constant harmonic function $f \in L^1(M)$? Who can answer me or give me a counter example? Thank you very much!
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1answer
170 views

The extension of diffeomorphism

Let ${\Omega _1}$,${\Omega _2}$ be two open sets in $\mathbb R^n$ and $f$ is a diffeomorphism between them. For every $x$ in ${\Omega _1}$, is there an open set $\Omega_{x} \subset \Omega_1$ and a ...
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215 views

Dimension of the space of matrices with constant determinant.

I'm looking for the dimension of the space of $n\times n$ real matrices $A$ such that $\det(A)=c$. I apply 2 different approaches and I get different answers. which one is correct? 1) So we ...
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544 views

Sufficient Conditions for Ricci Tensor to be Diagonal

What are the strongest (or most useful) conditions on a metric for it's Ricci tensor to be diagonal? I've read that if the metric is explicitly dependent on only one variable then the Ricci Tensor is ...
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115 views

Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
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1answer
74 views

Variations in a Riemannian Manifold

Let be $M$ a Riemannian manifold and $X,Y$ vector fields over $M.$ Now take $p\in M$ arbitrarily, my question is, how construc a variation $f:U\to M,$ $$U\subset ...
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1answer
166 views

Does the value of the covariant derivative at a point of the metric tensor depend only on the involved tangent vectors?

Let $\nabla$ be an affine connection on a pseudo-Riemannian manifold $(M,g)$. Let $c:[0,1] \rightarrow M$ be a differentiable curve and consider vector fields $Y,Z$ along $c$. Is it true that the ...
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1answer
113 views

$C_0$ Convergence of Metrics under Ricci Flow when $\chi(M) = 0$.

I am a beginning graduate student, and I am trying to learn basic aspects of Ricci Flow via the uniformization theorem for compact surfaces. I am reading Chow and Knopf's introductory book as well as ...
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2answers
129 views

What is the limit distance to the base function if offset curve is a function too?

I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
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Set of points where an application has rank $m$ is a smooth manifold.

Can someone help me with this problem? I have a $C^1$ function $G\colon\mathbb{R}^n\rightarrow \mathbb{R}^m$, where $k=n-m> 0$. If $M$ is the set of points $x\in G^{-1}(0)$ such that $(DG)_x$ has ...
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633 views

Variety vs. Manifold

In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am ...
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1k views

How to find the tangent space to a matrix space

I have a hard time approaching these types of problems. In an article it had claimed that the tangent space to all symmetric matrices with the same signature as $M$ at a matrix $M$ is the set of all ...
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176 views

The Closed disc $D$ is a manifold with boundary

It is obvious geometrically, but how one proves with a few words, analytically, the statement above?Additionally, if one has a smaller open disc $D_\epsilon$ of radius $\epsilon$ centered at a point ...
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1answer
554 views

Projective Space orientation

I'm trying to prove that the projective plane $\mathbb{P}^n$ is orientable is and only if $n$ is odd. To do that that, I have a hint,to prove that the antipodal map is orientation preserving if only ...
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Complete a set of functions to obtain a system.

Let $M^m$ be a $m-$manifold and $C=\{y^1,\dots,y^k:U\subset M\rightarrow \mathbb{R}\},\;k<m,$ a collection of functions such that on a point $p$, we have that $dy^1|_p,\dots,dy^k|_p$ is linearly ...
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380 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
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1answer
108 views

trouble with understanding notation - partition of unity, section

I am currently working with a book ("Fourier Integral Operators" by J.J. Duistermaat) that mentions a differential geometric construction that I struggle to understand. Here is the setting: Suppose ...
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382 views

Characterization of gradient vector fields

Let $V$ be a vector field on a smooth manifold $M$. Are there nice conditions under which there exists a (Riemannian) metric on $M$ such that $V$ is the gradient of some smooth function on $M$? One ...
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1answer
191 views

A question about orientation on a Manifold

Let $(U,x_1,x_2,\ldots , x_n)$ be a chart for a orientable manifold $M$, why $(U,-x_1,x_2,\ldots , x_n)$ is a chart for the Manifold $-M$, the same manifold with reversed orientation?
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111 views

Curves on a circle

Is it possible at every point $p=(x,y)$ on the unit circle, there is a continuous curve $C_p$ passing through it, a curve which is not only the single point $p$, and all these curves are pairwise ...
2
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1answer
135 views

Avoiding rationals $\implies$ constant

If $f:(0,1)\rightarrow\mathbb{R}^n$, with $n>1$, is a continuous curve in $\mathbb{R}^n$, with $f(p)=(x_1(p),x_2(p),...,x_n(p))$. Must the set of $p$ such that for some $i$, $x_i(p)$ is a rational ...
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261 views

Change of variables formula

Consider an example. Let $f(x)$ be a function on the unit sphere $S^{n-1}$. Consider an integral $$ \int\limits_{S^{n-1}} f(x) \, dx $$ I want to make a substitution $x = x_{0}t + \sqrt{1-t^2}y$, ...
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Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that i have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
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The cone is not immersed in $\mathbb{R}^3$

The problem is to show that the cone $ z^2= x^2+y^2$ is not an immersed smooth manifold in $\mathbb{R}^3$.
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Curves and first fundamental form

Would I be right to think that if I have a coordinate system $(x,y)$ so that the lines/curves where one coordinate is fixed, so something like $x=a$ and $y=b$, always intersect at the same angle, then ...
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155 views

Why topological stratification is useful?

My main focus is on the applications of stratification in complex/abstract Algebraic geometry especially, from the scheme-theoretic viewpoint and (Added) Moduli spaces. I have a vague feeling that ...
3
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1answer
184 views

Coordinate change for metrics

I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me. As far as my understanding goes, given a Riemannian metric ...
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3answers
368 views

Extending geodesics to vector fields

Let $c$ be a geodesic on a Manifold $M$. Some books define $c$ to be a Geodesic iff $\nabla_{c'}c'=0$. Therefore for every $c(t)$ the Geodesic must be extendable into a smooth vector field on an open ...
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moving a basis along a curve (parallel transport)

I'm considering a Riemannian Manifold $M^m$ and a Basis $\{X_1 ,...,X_n\}$ of the tangent space $T_pM$. When I consider now the parallel transport $E_i$ of the vectors $X_i$ along a curve c, then the ...