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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Are truth tables a valid method to prove an iff statement?

I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if ...
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14 views

normal curvature2

Question:If the surface $S_1$ intersects the surface $S_2$ along the regular curve $C$, then the curvature $k$ of $C$ at $p \in C$ is given by $k^2\sin^2\theta = \lambda^2_1 + \lambda^2_2 - ...
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23 views

Showing the product rule for the Euclidean connection wrt the Euclidean metric.

I'm confused about a few things in Lee's book on Riemannian geometry. On page 67, Lee writes that it is easy to compute the following in terms of the standard basis where $\overline{\nabla}$ is the ...
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15 views

Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermition form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...
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28 views

Normal curvature

I'm working on a problem but I don't understand some parts to. I get why $|\sin \theta|= |N_1 \times N_2|$ but why does $|N_1 \times N_2|= |n \times (N_1 \times N_2)|$. Is there a formula do this that ...
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18 views

Normal form of the Frenet Matrix

A book that I am going through states that on reducing to normal form the Frenet matrix in the Frenet-Serret formula, one ends up with the matrix: $$ K= \begin{bmatrix} 0 & ...
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24 views

parallel transport_covariant derrivative

Let $u(s,t) $ be smooth and $X(t) $ vector field along $u(s_0,t) $. Denote with $P_s= P_s( u(s,t), u(s_0,t))$ parallel transport in $s$ direction along $u(s,t) $ from $u(s_0,t) $ to $u(s,t) $. Does ...
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1answer
31 views

Transverse intersection in a compact manifold

Is it true that if $M$ is a compact manifold and $X,Y$ are submanifolds of $M$ which intersect transversely that the intersection $X\cap Y$ consists of finitely many points? I'm trying to understand ...
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52 views

Question about Hopf-Rinow theorem

I'm studying Hopf-Rinow theorem and I don't see a step in the proof. Could someone help me, please? (Definition) Let's $(M, \langle,\rangle)$ an ANII(axiom numerability 2) and Hausdorff Riemannian ...
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84 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
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33 views

Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
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19 views

Generalisation of the concept of solid angles in curved space

I wonder if there are standard extensions of the concept of solid angle in curved space. There should be because physicists who study radiation in curved space would run into this kind of needs. If ...
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36 views

Curvatures of 4-Dimensional Parallel Curve

Let $\vec{\alpha}: I \rightarrow \mathbb{R}^4$ be an arc-length parametrized curve in $\mathbb{R}^4$ with curvatures $k_1, k_2, k_3$. The principal normal unit vector is $\vec{n} = ...
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0answers
26 views

How do I Prove that $M:=A(X_u)\cdot X_v=A(X_v)\cdot X_u$?

How can we show that $$M:=A(X_u)\cdot X_v=A(X_v)\cdot X_u\ ,$$ where $A$ is the shape operator and $X_u$, $X_v$ are the coordinate vectors?
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1answer
88 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. ...
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What is wrong with this exercise in do Carmo's Differential Geometry?

This is an exercise in do Carmo's Differential Geometry: Let $\alpha : I \longrightarrow S$ be a curve parametrized by arc length $s$, with nonzero curvature. Consider the parametrized surface ...
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3answers
82 views

Manifolds and their dimension

Why are the following two sets manifolds and what are their dimensions? The set of all 2 by 2 matrices with determinant 1. The set of all inner products on $\mathbb R^3$. I have difficulty in ...
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Lateral area of a cone with oblique base (elliptical base)

Can someone help me find the lateral area of a straight circular cone with oblique base ( inclined elliptical base) as seen at the following link. ...
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0answers
73 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
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1answer
27 views

Condition that a homeomorphism is an isometry

The statement that every isometry is a homeomorphism is true (correct me if I am wrong). What is the condition for any homeomorphism is an isometry? In particular, is it true that any homeomorphism ...
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47 views

hairy ball theorem?

I was reading an article on the hairy ball theorem, and found this: every zero of a vector field has a (non-zero) "index", and it can be shown that the sum of all of the indices at all of the ...
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19 views

absolute value of torsion of asymptotic curve

Question:Prove the the absolute value of the torsion $\tau$ at a point of an asymptotic curve, whose curvature in nowhere zero, is given by $|\tau| = \sqrt{-K},$ where $K$ is the Gaussian curvature ...
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1answer
19 views

How to parameterized implicit curve.

How could I parameterize: $$\frac{1}{2}\left(x^2+y^2\right)-\frac{1}{3}x^3=\frac{1}{6}$$ as $x(t)$ and $y(t)$?
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43 views

compact surface of revolution

I have to prove that a compact surface of revolution is diffeomorphic to a sphere or to a torus. And show that $\int_{S} K dA= $ =$\{4\pi,$ if S is spherical type 0, if S is toric type $\}$, ...
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let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
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0answers
44 views

Completeness of a Riemannian manifold with boundary

I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined ...
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2answers
88 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
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18 views

Any Straight Line Contained in a Surface is Asymptotic and Hyperbolic “Squares”

I have to prove that "any straight line $\alpha$ contained on a surface $S$ is an asymptotic curve and geodesic (modulo parametrization) of that surface $S$". Can I have hints at tackling this ...
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2answers
73 views

Are homeomorphic differentiable manifolds actually diffeomorphic?

Let $M$ and $N$ be two n-dimensional smooth manifolds.Suppose their underlying topological spaces are homeomorphic through $f$. Does $f$ automatically become a diffeomorphism with respect to the given ...
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1answer
41 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
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36 views

Christoffel symbols vanish in a system of normal coordinates.

I'm reviewing for a differential geometry exam and am getting stuck in a proof. This is based on question 4 from section 4-6 from little Do Carmo. Show that in a system of normal coordinates ...
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33 views

Structure of level sets of a noncritical point of a smooth function on a two dimensional domain

Let $\psi$ be a smooth function on a two dimensional simply connected domain $\Omega$ such that $\psi=0$ on the boundary $\partial \Omega$. Suppose $\rho$ is not a critical value of $\psi$ then it is ...
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1answer
26 views

2 dimensional Laplace's equation in polar coordinates

The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdot u ...
2
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1answer
42 views

Finding relationship between Laplace-Beltrami operators of two spheres

Let $S$ and $T$ be spheres with radius $R_S$ and $R_T$ respectively. Define the diffeomorphism $\Phi:S \to T$ by $\Phi(s) = \frac{R_T}{R_S}s$. Given a function $u:T \to \mathbb{R}$, we can define ...
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1answer
26 views

Tangents of a Curve lie on a Cone

Prove that if all tangent vectors to the curve $α(t) = (3t, 3t^2, 2t^3)$ are drawn from the origin, then their endpoints are on the surface of a circular cone with the axis the line $x − z = y = 0$. ...
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1answer
19 views

Confusion about $\Delta_{S^n} u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$, why do we need to divide by $|x|$?

Let $S$ be a sphere of radius 1. We know the formula $$\Delta_S u(x) = \Delta u\bigg(\frac{x}{|x|}\bigg)$$ holds for a function $u:S \to \mathbb{R}$ where $\Delta_S$ is the Laplace-Beltrami and the ...
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1answer
45 views

Link between a topological space and a manifold

A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that, (i) any union of these subsets is one of the subsets. (ii) any ...
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0answers
24 views

How to decompose connections on the complexified orthonormal frame bundle?

Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)}(E)$ the orthonormal frame bundle. I say that $P^{c}:=F_{SO(n)}(E)\times_{SO(n)} ...
2
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1answer
39 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
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0answers
42 views

A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
1
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2answers
68 views

Proving a space is a manifold

Given a topological space defined as $A=A_1 \cup A_2$ with $A_1=\{(x,y) \in R^2 \space \space|\space \space x^2+y^2=1, x<0\}$, $A_2=\{(x,y) \in R^2 \space \space|\space \space |x|+|y|=1, ...
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0answers
23 views

How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
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1answer
93 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
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2answers
84 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
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1answer
56 views

Help Finding Orthonormal Frame and Coframe Based on First Fundamental Form

Given the metric (for $x^2<1$) $g=dx^2+2x dxdy+dy^2$ (in first fundamental form) I'm trying to find the orthonormal frame and its coframe I have found (I think) the orthonormal frame to be ...
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0answers
48 views

How can we extend the proof that Lie groups are orientable to parallelizable manifolds?

Let $G$ be a Lie group with identity element $e$ and dimension $n$. I know we can take a non-vanishing $n$-form (call it $\omega$) on the vector space $T_eG$. For a Lie group, as we have an ...
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1answer
26 views

Prove that function F is diffeomorphism

I have a question in the book 'Elementary differential geometry' Prove that if a one-to-one and onto mapping $$F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$$ is regular, then it is diffeomorphism. I ...
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1answer
32 views

How to find a circle which approximates an unit speed curve at the origin?

A parametrization of a circle by arc length may be written as $$\gamma(t)=c+r\cos(t/r)e_1+r\sin(t/r)e_2.$$ Suppose $\beta$ is an unit speed curve such that its curvature $\kappa$ satisfies ...
2
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1answer
34 views

Reparametrize the curve problem in Differential geometry

Reparametrize the curve $\alpha(t)=(e^{t},e^{-t},\sqrt{2}t), \; \alpha: \mathbb{R} \rightarrow \mathbb{R}^{3}$, using $h(s)= \log(s)$ on $J:s>0$. Check the equation in Lemma in this case by ...
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1answer
41 views

Curve problem in Differential geometry

Find the coordinate functions of the curve $\beta = \alpha(h)$, where $x$ is the curve in $$\alpha (t)= \left( 1+\cos t, \, \sin t, \, 2\sin \frac{t}{2} \right)$$ for all $t$ and $h(s)=\cos^{-1}(s)$ ...