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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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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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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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 $t\in S^1$. ...
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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 ...
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712 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 ...
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387 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 ...
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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 ...
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2answers
627 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
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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 ...
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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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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 ...
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295 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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125 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 ...
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1answer
214 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
93 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!
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445 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 ...
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114 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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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87 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 ...
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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 ...
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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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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 ...
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3answers
159 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 ...
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224 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 ...
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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 ...
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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 $G$. ...
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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 ...
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1answer
499 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 ...
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318 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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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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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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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 ...
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128 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 ...
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1answer
154 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 ...
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1answer
69 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. ...
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1answer
225 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 ...
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1answer
55 views

Distribution and Tangent Bundle

Let $F=\{F_p; p\in M\}\subseteq TM$ be a rank $k$ smooth distribution. Can anyone explain-me what is the set $$\displaystyle\nu(F)=\frac{T_pM}{F_p}.$$
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Vector Bundle Doubt..

Well I have a doubt about a rank $k$ vector bundle. My definition of vector bundle is: A rank $k$ vector bundle is a triple $(\pi, E, M)$ where $E$ and $M$ are smooth manifolds and $\pi:E\rightarrow ...
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66 views

Lie bracket in local coordinates.

$\bf 14.9.$ Lie bracket in local coordinates Consider the two vector fields $X,Y$ on $\mathbb{R}^n$: $$X=\sum a^i\dfrac\partial{\partial x^i},\qquad Y=\sum b^j\dfrac\partial{\partial x^j},$$ where ...
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68 views

Why $d(f(x,y)dx)=df(x,y) \wedge dx$

Who can tell me why $d(f(x,y)dx)=df(x,y) \wedge dx$?
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81 views

Show the regular submanifold

Please help me how sdo I show such a problem? I Will be happy to teach me. Thank you
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Distances of points on geodesics

The setting: -Let $(M,g)$ be a complete Riemannian manifold and let $\pi:E \rightarrow M$ be its universal covering with the pullback metric. -Let $\alpha,\beta:[0,1] \rightarrow E $ be two minimal ...
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1answer
223 views

Classification of flat complex line bundles

I'm having a contradiction with two different classifications of flat complex line bundles over a manifold $X$. Suppose for simplicity that $H^2(X;\mathbb Z) = 0 = H^1(X;\mathbb C)$. Then the only ...
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70 views

Newman-Penrose tetrad questions

I have a question/exercise relevant to students of mathematical relativity: Let $\left \{ l^{a},n^{a},m^{a},\bar{m}^{a} \right \}$ be a Newman-Penrose tetrad, where only the direction of $l^{a}$ is ...
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1answer
62 views

Lagrangian subspaces

Let $\Lambda_{n}$ be the set of all Lagrangian subspaces of $C^{n}$, and $P\in \Lambda_{n}$. Put $U_{P} = \{Q\in \Lambda_{n} : Q\cap (iP)=0\}$. There is an assertion that the set $U_{P}$ is ...