A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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Heat Equation on Manifold

Laplacian operator is defined well on Riemannian manifold, denoted by $\Delta$. Therefore people can study PDE $\Delta f=0$ on manifold. So is there any analogy to heat equation or wave equation on ...
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Basic misunderstanding of the theorema egregium

The theorema egregium demonstrates that the Gaussian curvature, $K$, is an intrinsic property. What I think this means is that if you know the metric corresponding to the surface, then you can compute ...
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Computing distance on a sphere

Let's say I want to compute the distance between two far points on Earth, say Toronto and Brazil. I can do this by getting in my car, setting my odometer to zero and then driving to Brazil. For me, ...
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for a compact manifold $M$, is the dual space of $H^1(M)$ equal to $H^{-1}(M)$?

Let $M$ be a compact Riemannian manifold. Is it true that $$(H^1(M))^* = H^{-1}(M)?$$ is there some intuitive explanation why? Or some reference? Thanks Here $H^1$ is the usual Sobolev space of $u ...
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Conditions on the metric function in a flat manifold

It is well known that a manifold is flat iff its Riemann tensor vanishes identically. However, the equation $R^{\mu}_{\nu\rho\sigma}=0$ is a differential equation for the metric tensor $g_{\mu \nu}$. ...
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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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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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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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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 ...
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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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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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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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328 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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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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$(n-1)$-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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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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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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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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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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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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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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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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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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Norm of the gradient of function $f$ 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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71 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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59 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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152 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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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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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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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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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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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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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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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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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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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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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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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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33 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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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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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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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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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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152 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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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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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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(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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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 ...