(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

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Integration by parts (Differential Geometry)

I am studying the proof of a theorem and I am stuck. It says that by integration by parts we get that: For $g(t)$ a variation of Riemannian metrics wih $g'(0)=h,$ $$\int_{M} (-\Delta (tr h) + ...
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26 views

A question about two parallel integrable distributions

Give a Riemannian manifolds $(M,g)$,$\nabla$ is its connection. Suppose we have two distributions $E$ and $F$ on $(M,g)$,that are orthogonal complements of each other in $TM$.In addition,assume that ...
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17 views

Does any chaotic operator $T$ in infinite-dimensional Hilbert space is an isomorphism?

Suppose that any operator is an "Isomorphism" if it is mapping a hard equation to easier algebraic equations. Can I take this as a necessary criteria to judge that: "Any chaotic operator $T$ in ...
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1answer
37 views

Formal adjoint of divergence

We define the so-called conformal Killing operator $K$ mapping (1,0) vectors to (0,2) tensors by $$K(X)_{ab} = \frac{1}{2}\nabla_aX_b+ \nabla_bX_a -\frac{2}{3}(\text{div}X) g_{ab}.$$ Here $g$ is the ...
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52 views

Parallel hypersurfaces in a riemannian manifold and focal points

For $M^n$ a riemannian manifold and $S$ a hypersurface, if we consider $S_t=\{exp^\perp(v):v\in T(S)^\perp,|v|=t\}$ and $f_t:S\rightarrow S_t:p\mapsto exp^\perp(t\eta)$ with $\eta$ the unit normal ...
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33 views

Christoffel Symbols on a Surface

In Do Carmo's Differential Geometry of Curves and Surfaces he does the following: Let $\vec r$ be a parametrization of a surface $S\subset\mathbb{R}^3$ so that $\vec r_u,\vec r_v$ forms a basis for ...
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2answers
49 views

Hermitian manifold counterexample

I'm trying to come to come to grips with the notion of a hermitian manifold. Although I know some examples of hermitian manifolds, I am more interested in counterexamples: naturally occurring ...
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36 views

Surfaces on which not every pair of points is connected by a geodesic

Let $S$ be a surface in $\mathbb{R}^3$. I believe that, if $S$ is smooth, bounded, and closed, then, for every pair of points $x,y \in S$, there is at least one geodesic $\gamma$ connecting $x$ to ...
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90 views

Hodge star isomorphism

In Petersen's Riemannian geometry text, he defines the Hodge operator $*: \Omega^k(M) \to \Omega^{n-k} (M)$ in the standard way. He then proves (Lemma 26, Chap 7) that $*^2: \Omega^k(M) \to ...
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90 views

Prove: $(\delta^\nabla\text{d}^\nabla+\text{d}^\nabla\delta^\nabla)h=\nabla^*\nabla h-\mathring{R}_gh+h\circ\text{Ricc}_g$

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and adjoint $\nabla^*$, and exterior derivative $\text{d}^\nabla$ and adjoint $\delta^\nabla$. For a symmetric 2-covariant ...
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45 views

$(n-1)$-alternative tensor on E are decomposable

$E$ is a real vector space with dimension $n$ and $E^*$ is dual space of $E$. Assume $\alpha \in Λ^{n-1}(E)$ Show that there exists $\alpha_1,\alpha_2,...,\alpha_{n-1} \in E^*$ such that ...
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46 views

double Hodge star operator

$*$ is the Hodge star operator acting on differential $k$-forms on $\mathbb{R}^{n}$ as below: $$*:Λ^k(\mathbb{R}^{n}) \to Λ^{n-k}(\mathbb{R}^{n})$$ $$\alpha \wedge (\star \beta) = \langle ...
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1answer
37 views

Discrete faithful representation in $PSL(2,\mathbb R)$ and horocycles in hyperbolic space

Let $S$ be a closed oriented surface of genus $g>1$. Is the following true ? Let $\alpha,\beta\in \pi_1(S)\backslash \{1\}$ and $\rho:\pi_1(S)\rightarrow PSL(2,\mathbb R)$ be a discrete ...
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29 views

constant positive K surface

Hilbert's Theorem states that there exists no complete analytic (class Cω) regular surface in $R^3$ of constant negative Gaussian curvature K. For positive Gaussian curvature also when the sphere and ...
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31 views

geodesics, covering map and its lift

Followings are given problems. Let $f:(M,g)\rightarrow (N,h)$ a covering map that is a local isometry, and let $p\in M$. If $\gamma:[0,1] \rightarrow N$ is a geodesic such that ...
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53 views

On the definitions of $n$-manifold etc.

I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about ...
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125 views

Does Nash's theorem allow an embedded representation of the Riemann tensor without loss of generality?

Does Nash's theorem allow an embedded representation of the Riemann tensor without loss of generality? Based on what is found here Nash embedding theorem: "The Nash embedding theorems (or imbedding ...
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1answer
28 views

Prove without coordinates that covariant derivatives are “almost” related under isometric immersion?

I'm trying to solve this problem: Let $F : (M,g) \to (N,h)$ be an isometric immersion. For any $p \in M$, let $\pi_p$ be the orthogonal projection from $T_{F(p)}N$ to the image of $dF_p : T_pM \to ...
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90 views

frustrating experience about differential geometry

I am felling rather frustrated now, after taking a long time to study differential geometry, but with little progress... Indeed my major is mainly numerical analysis. I am studying modern geometry, ...
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17 views

norm of $\text{grad} f$ on manifold $g(x)=c$

Let $g,f:\mathbb{R}^2 \to \mathbb{R}$, $M=g^{-1}(c)$. Let say that we manage to write $f(x,y)=f_{*}(x)$ for $x\in M$. When I was calculating the square of norm of $$\nabla f_M (x,y)=\nabla ...
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43 views

A question about incompressible vector field

Let $ X $ be a unit vector field on $\mathbb{R}^2$,with canonical metric $g$ and connection $\nabla$ .Show that if $$divX=0$$ then $$\nabla X=0$$ I tried: Let $X=(f,g)$,$Y=(-g,f,)$,then $X,Y$ is ...
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19 views

Product of Riemannian manifold and product metric

according to wikipedia the product metric between 2 metrics is the metric given by: $d(x,y)=\sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$ Now if $(M,g_m)$ and $(N,g_n)$ are 2 Riemannian manifolds we can ...
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26 views

Riemannian metric and its manifolds' completeness

Consider the Riemannian metric $g$ in $\mathbb{R}^2$. What is the condition for $g$ to be the Riemannian manifold $(\mathbb{R}^2, g)$ becomes complete? If the above question solved, can i simply ...
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43 views

Riemann sphere, metric derivation-Completed

I have been calculated Riemann sphere, but i got stuck with calculating its metric. Consider complex plane $\mathbf{C}$ and its point $\zeta=\xi+i\eta$. And consider a point in $S^2 / (0,0,1)$ which ...
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2answers
46 views

Proving that a curve is a Geodesic in the Poincaré Half-Plane

Let $\mathbb{H}^2$ be the Poincaré Half-Plane, that is, $\mathbb{R}\times \mathbb{R}_+^*$ with the Riemannian metric $$\langle u,v \rangle_{(x,y)} = \frac{u \cdot v}{y^2}$$ I was asked (in a test) to ...
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40 views

The existence of complete Riemannian metric

If $M$ is a differential manifold, can we necessarily find a complete Riemannian manifold on $M$? (I know we can find a Riemannian metric without completeness assumption.)
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20 views

Minimal surface between two non coaxial rings

I'm currently studying minimal surfaces using the Euler-Lagrange equation. I'm particularly interested in minimal surfaces between two circles. I have already examined the case of two coaxial ...
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39 views

Poincare disk and mobius transformation

I have following problem Consider Poincare disk. $i.e$ $\mathbb{M}=\{ (x,y) \in \mathbb{R}^2 : x^2+y^2 <1 \}$ with metric $ g= 4\frac{dx^2 +dy^2}{(1-x^2-y^2)^2}$ Show the complex mobius ...
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63 views

computation on hyper surface $z=x^2+y^2$

I have problem with following exercise Consider the hypersurface $M$ parametrized by $z=x^2+y^2$. Endow this with the Riemannian metric induced from the $\mathbb{R}^3$. Compute the sectional ...
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51 views

Question about parallel fields and geodesics

Suppose $V$ is a vector fields on a geodesic $\alpha$. Show that $V$ is parallel if, and only if $\| V\| $ is constant and the angle between $V$ and $\alpha'$ is constant. I have done the following: ...
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1answer
88 views

Expressions with the connection form in a Riemannian manifold $M^2$.

Let $M$ be a $2$-dimensional Riemannian manifold, and ${\bf x}: U \subset \Bbb R^2 \to M$ be a parametrization of $M$. Suppose that $\bf x$ is orthogonal, that is, $F = \langle {\bf x}_u,{\bf ...
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34 views

Can you give a example about of curvature tensor

Can your give a Riemann manifold $(M^n,g)$,let $R(X,Y,Z,W)=g(R(Z,W)X,Y)$,and under some coframe $w^1,...w^n$, $$R=R_{ijkl}w^i \bigotimes w^j\bigotimes w^k \bigotimes w^l$$ such that,$\forall i,j ...
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43 views

Uniqueness of smooth/symplectic/etc structure

It is well-known that every topological manifold $M$ of dimension $\le 3$ admits a unique smooth structure. That is to say for any choice of atlas on $M$ like $A$ and $B$, the smooth manifolds $(M, ...
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1answer
35 views

Proper definite of riemann integral (limit version)

I am sort of confused. Suppose we are given the series, $\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n} \frac{k^{99}}{n^{100}}$ How can this be written as an integral, and what would the variable ...
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1answer
35 views

Area in the Hyperbolic Plane

Let $D_2(0) = \{(x,y) \in \mathbb{R}^2 \ | x^2+y^2 \leq 4\}$ with the Riemannian metric $$\langle u,v\rangle_{(x,y)} = \frac{u\cdot v}{g^2(x,y)} \ \ \ \ \ g(x,y) = 1 - \frac{x^2+y^2}{4}$$ I want to ...
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34 views

Showing the right half of the unit hyperbola is a complete metric space.

Let $f : \mathbb{R} \rightarrow \mathbb{R}^2$ be given as follows. $$f(\theta) = (\cosh \theta, \sinh \theta)$$ I want to argue that $\mathrm{im}(f)$ is a complete metric space with respect to the ...
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18 views

Metric for subset of $\mathbb{R}^n$

Consider the collection of points in $\mathbb{R}^n$ whose coordinates are all strictly positive. I want to think of this subset as a Riemannian manifold. Does anyone happen to know what the Reimannian ...
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30 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
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17 views

Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
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36 views

If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
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Riemann curvature product metric

Suppose that $M=M_1 \times M_2,$ with the product metric $g= g_1 \oplus g_2.$ Let $p\in M$ and suppose that $X \in T_pM_1$ and $Y\in T_pM_2.$ I want to show that $R(X,Y,Y,X)=0,$ at the point $p.$ I ...
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62 views

Topologies in a Riemannian Manifold

I'm studying Differential Manifolds using Manfredo do Carmo's Book (Riemannian Geometry) and although I see no mention of this in Do Carmo's book, it's really easy to see a Riemannian Manifold as a ...
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How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
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42 views

A simple metric question

In their article Killing vector fields of standard static spacetimes, Dobarro and Unal derived the following simple identity. Note that if $h:I→R$ is smooth and $Y,Z∈ {\frak{X}}\left(I\right)$, ...
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42 views

Existence of normal orthogonal frame on sphere such that it is normal at every point in a neighbourhood?

It seems that on sphere $S^{n-1}$, there exists a better frame than the usually normal frame. In the literature, some author asserts that there exists a local orthogonal frame ...
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38 views

The uniqueness of the Einstein metric on $\mathbb CP^n$

Is the Fubini-Study metric the unique Einstein metric (up to scaling by a constant) on $\mathbb CP^n$? More restrictively, Is the Fubini-Study metric the unique Kahler-Einstein metric (up to scaling ...
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39 views

Prove that there are no complete regular minimal surfaces lying above a paraboloid

Prove that there are no complete regular minimal surfaces lying above a paraboloid contained in $U=\{(x,y,z) \in \mathbb{R}^3 : a(x^2+y^2)<z\}$. Here $a>0$. I've had this problem on my mind ...
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32 views

Can you define a projection with out inner product

Does a projection require a concept of inner-product?
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67 views

(locally) “almost convex” property of the distance function in a general Riemannian manifold

Given two constant-speed geodesics $\gamma_1$ and $\gamma_2$ in an euclidean space $\mathbb E^n$, it is possibile to see that: $$ t \mapsto d(\gamma_1(t), \gamma_2(t)) $$ is a convex function. The ...
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31 views

About Geodesic polar coordinate

What is different from geodesic polar coordinates and other polar coordinates? Geodesic polar coordinates has a form of $$ds^2=dr^2+f(r,\theta)^2\,\,d\theta^2$$ In $S^2$, $f(r,\theta)=\sin(r)$ which ...