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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354 views

Differential of smooth function on manifold

In the book I am using, the author defines differentials in the following way. Given smooth manifolds $M,N$ and a smooth mapping $\psi:M\to N$ define the differential $d\psi_m$ at a point $m\in M$ as ...
3
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1answer
145 views

Basic question about definition of Chern classes

Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)? If we define chern ...
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0answers
82 views

Cotangent bundle of a complex projective space

How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?
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1answer
299 views

Complete non-vanishing vector field

Let $M$ be a non-compact smooth manifold. Suppose we have a nowhere-vanishing smooth vector field X. Is this vector field complete? I know it is when $M$ is compact. However, I am unsure in the ...
0
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1answer
306 views

Surfaces of Constant Gaussian Curvature

I'm preparing for an exam and I would like to know what are some examples of surfaces with constant Gaussian curvature such as surfaces with $k=0, \pm1$
3
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0answers
226 views

Submanifold with boundary of a manifold with boundary

Let $M$ be a smooth manifold. (1) A subset $S$ of $M$ that with the subspace topology is a topological manifold (with or without boundary), together with a differential structure that makes the ...
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1answer
297 views

Infinite dimensional constant rank theorem

Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
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0answers
54 views

How prove prove the identity $J_{t}=J\triangledown\cdot V$

prove the identity $$J_{t}=J\triangledown\cdot V$$ where $J=\begin{vmatrix} x_{u}&x_{v}\\ y_{u}&y_{v} \end{vmatrix}$ is the Jacobian of the map $x(u,v;t)\in R^2$ and $$V(x)=x_{t}$$
6
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2answers
407 views

Different types of domains $\Omega \subset \mathbb{R}^n$ in PDEs

In PDEs I often read things like: Let $\Omega$ be a bounded Lipschitz or $C^1$ or $C^2$ or $C^\infty$ domain But I have no clue what this means in real life. I understand ...
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1answer
528 views

Implicit Function Theorem and Rank Theorem Misunderstandings.

Regular values are useful because of the generalization of the first part of the implicit function theorem: if $q$ is a regular value of $f:M \to N$ (with dimension $m$ and $n$ respectively), then $$A ...
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0answers
57 views

coordinate transformation of the local pull back of the Maurer Cartan form

This questions asks how to express the pull back of the Mauere Cartan form on a Lie group to a smooth manifold. The Lie group,$G$, is the structure group of a smooth vector bundle over the ...
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1answer
79 views

Orientability of $P_{\bf R}T{\bf RP}^{2n}$

I know the following fact : (1) $ {\bf RP}^{2n}$ is non-orientable. (2) $ {\bf RP}^{2n-1}$ is orientable. (3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable. (4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
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1answer
59 views

Question about a specific case of the argument principle for maps of circles.

Problem Statement: Let $f:S^1\rightarrow S^1$ be a smooth map of manifolds where $S^1=\frac{[0,1]}{0~1}$, and let $f'(t)\in \mathbb{R}$ be given by the $df_t[1]_t=f'(t)[1]_{f(t)}$ at each $t\in S^1$. ...
3
votes
2answers
160 views

Extending a smooth map

When can I extend a smooth map $f:\mathbb{R^2}-\lbrace 0 \rbrace \to S^1$ to a smooth map $\tilde{f}:\mathbb{R^2} \to S^1$. For instance, consider $g(x,y)=(x,y)/\sqrt{x^2+y^2}$? Am I able to extend ...
2
votes
1answer
724 views

Gradient in Riemannian manifold

I have a calculation involving a gradient and a parametrization, but I haven't been able to find out the relation between them. Let me explain. Let $f:X↦R$ be a smooth function and $\mathrm{grad}f\in ...
5
votes
0answers
391 views

Second derivative of a metric in terms of the Riemann curvature tensor.

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial ...
2
votes
1answer
91 views

Show that there is no surjective smooth function $S^1 \to S^1\times S^1\times S^1$

This is not homework, but a sample test question. The question is: Show that there is no surjective smooth function $$S^1 \to S^1 \times S^1 \times S^1.$$ Now I can see that, for example ...
0
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2answers
629 views

what will be the parameterization of cone

I have question, I need more idea. can any one answer my question I have tried but i didnot get full idea I know this question we have to use parameterization of cone which i donot know in this case ...
2
votes
1answer
48 views

what is the inner product appeared in front of the integral?

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 ...
1
vote
1answer
126 views

Tangent Vectors in a Surface

As of recent, I've been studying Differential Geometry per the Dover Publication on the subject, and I've ran into a bit of an issue with tangent vectors to a parametric surface $ \mathbf{x}(u^1,u^2) ...
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1answer
125 views

Is a parameterization defined to be surjective and/or injective?

A parameterization is a mapping used in differential geometry for describing a manifold, and in statistics for describing a family of distributions, and may be used for other applications I don't know ...
3
votes
1answer
298 views

Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.

Given a local diffeomorphism $f: N \to M$ with $M$ orientable. Why is $N$ orientable? My professor wrote this in class without giving a proof and said "you should try to prove this for fun :)". I ...
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1answer
126 views

geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
1
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1answer
216 views

Is this distribution involutive?

For two days I've been trying to show the following: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and consider the smooth distribution $$F=\{F_p=DR_p(e)\mathfrak{h}; p\in G\},$$ where ...
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1answer
94 views

Meaning of modulo diffeomorphism

I faced this sentence: we consider the space of Riemannian metrics modulo diffeomorphism and scaling. Can anyone explain to me what is the meaning of modulo diffeomorphism and scaling? Thanks!
2
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1answer
450 views

Holonomy of the sphere

I saw an example in which the holonomy of $\mathbb{S}^n$ with the standard metric is calculated. I'm just starting to get familiar with holonomy groups and I wanted to know what could one do by ...
2
votes
0answers
115 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
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1answer
100 views

Question about lie bracket..

Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by ...
0
votes
1answer
1k views

what is the right circular of cone and what is the right circular of cylinder

I have some questions. 1)what is the parametrization of cone and what is the parametrization of cylinder? 2) what is the right circular of cone and what is the right circular of cylinder? I ...
0
votes
1answer
200 views

Find the differential equations that are satisfied by geodesics on the torus with parametrization given

I have question Find the differential equations that are satisfied by geodesics on the torus with parametrization given $X(u,v)=((R+rcos(u))cos(v),(R+rcos(u))sin(v),rsin(u))$? I hope someone can ...
2
votes
1answer
151 views

Trouble understanding differential forms. A basic question: what does $w \times dw$ mean?

After reading [1] and [2] I (kind of) understand what differential forms are, but I am still having trouble understanding the following argument from [3,Lem.4.2]: Let $\mathbb{T^3_n}$ be the ...
2
votes
1answer
156 views

Poisson bracket of coordinates

I just derived that in local coordinates (it suffices to centre) around $0$, that $$\{f,g\}(x)=\sum_{i,j}\{x^i,x^j\}\frac{\partial f}{\partial x^i}\frac{\partial g}{\partial x^j}$$ only using the ...
1
vote
1answer
88 views

Calculating Principal curves

I have been given a surface patch, $X(u,v)$, and I have calculated its unit normal, coefficiants of its first and second fundamental form and found its principal curvatures. Now it's asking me to find ...
4
votes
2answers
400 views

Motivation for the study of the Chern connection

Given a Hermitian metric $H$ over a holomorphic vector bundle $E$ with holomorphic structure $\overline{\partial}$, there exists a unique connection $\nabla$ (named afer Chern) satisying the following ...
2
votes
1answer
148 views

Computing $n$-th external power of standard simplectic form

I need some help: Define a 2-form on $R^n$ by $\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+...+dx_{2n-1}\wedge dx_{2n}$. How to compute $\omega^n:=\omega\wedge\omega\wedge\ldots\wedge\omega$?
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1answer
58 views

Computing the unit normal vector - Simplifying help

I have a surface $$X(u,v) = \left(3uv^2 - u^3 - \frac{u}{3}, 3u^2v - v^3 - \frac{v}{3}, 2uv \right), $$ and the cross product $$(X_u \times X_v) = \left(3(u^2 + v^2) \frac{1}{3} \right) \cdot ...
2
votes
3answers
160 views

symmetric positive definite matrices

Why must a symmetric positive definite matrix must be invertible? I'm reading a proof of the Levi-Civita theorem in differential geometry but the author states this without proof and I haven't been ...
8
votes
1answer
225 views

Triangulation of a manifold adapted to a submanifold

I am not extremely proficient in topology, and am concerned with the following question : Given a compact manifold $X$ and a submanifold $Y \subset X$, is it always possible to find a triangulation ...
2
votes
1answer
57 views

Geometric intuition for convexity of hypersurfaces in Riemannian manifolds.

I would like to get a geometric intuition behind convexity of hypersurfaces in Riemannian manifolds: Recall that a hypersurface in some Riemannian manifold is said to be convex, if its second ...
1
vote
1answer
80 views

Equality involving Lie Brackets

I have a question concerning Lie brackets: Consider the Lie bracket $$[, ]:\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g},$$ where $\mathfrak{g}=T_eG$ is the Lie algebra of a Lie group $G$. ...
3
votes
0answers
114 views

Tubular neighborhood with an additional projection

Let $i\colon L\to M$ be a submanifold inclusion. The tubular neighborhood theorem says that there is a tubular neighborhood of $i(L)$ in $M$ diffeomorphic to the normal bundle of $L$ in $M$, denoted ...
2
votes
1answer
505 views

Two results on the mean curvature of hypersurfaces

I am a physicist, now I consider a physically meaningful $N-1$ dimensional hypersurface $M^{N-1}$ embedding in the flat Euclidean space $R^{N}$. We have an explicit form of the hypersurface in the ...
0
votes
0answers
322 views

Show a cone isn't a regular surface

I know that a cone isn't a regular surface because I can't construct a chart with cts partial derivatives at its tip. But can anyone show me this last step rigorously? Why would any chart for the tip ...
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votes
1answer
71 views

Enneper's surface: differential geometry

Let $c \neq 0$ denote a real number. A surface patch is given as follows: $$ \alpha_c(u,v) =( \frac{u}{c^2} - \frac{u^3}{3} + uv^2,\frac{v}{c^2} - \frac{v^3}{3} + vu^2,\frac{u^2-v^2}{c} ) $$ where ...
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0answers
42 views

Do sections defined in different patches give the same element in an associated bundle?

We can read here in p. 10 (penultimate equation) that the sections on associated bundles (not necessarly vector bundle) are defined by functions $f:P\longrightarrow F$ that must satisfy the ...
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vote
1answer
178 views

Why do manifolds with negative sectional curvature not have conjugate points?

I'm trying to understand why manifolds with negative sectional curvature not have conjugate points. In fact for me it is sufficient to understand it for surfaces, but of course i'd be interested in ...
3
votes
0answers
129 views

Are geodesic flows on surfaces with negative curvature Anosov?

I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it. So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
2
votes
1answer
155 views

Bott connection

Can anyone help me showing the following: Let $E$ be a smooth vector bundle over $M$ and $F\subseteq TM$ a smooth distribution. A $F$-connection is a $\mathbb R$-bilinear aplication ...
2
votes
1answer
70 views

Positive curvature on holomorphic vector bundles

There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize: Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
2
votes
1answer
226 views

Prove that a tensor field is of type (1,2)

Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let $$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$ Prove that $N$ is a tensor field of ...