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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Geodesics on a 2-sphere

I've been doing some work where I need to find the geodesics in a given Riemannian Manifold. Let's take the example of the two sphere, for simplicity, with unitary radius. The distance between two ...
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376 views

How to derive covariant derivative and Lie derivative of tensors

1) As title says, how does one derive the following equation for covariant derivate of tensor: $A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$ where ...
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302 views

Intuition for Fisher information metric

In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\int \partial_i(\log ...
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190 views

Lie derivative of curvature

Let $M$ be a Kähler manifold, with Kähler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $\mathcal{L}_{X} g = 0$, where $\mathcal{L}_{X}$ is the Lie derivative along $X$. Let ...
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243 views

Maximum principle for minimal hypersurfaces

The well known local version of the maximum principle for minimal hypersurfaces asserts that if two minimal hypersurfaces $ M_1 $ and $ M_2 $ of $ R^n $ has a common point $ x_0 \in M_1 \cap M_2 $ ...
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240 views

A question about laplacian of the second fundamental form

Let $ f:M \rightarrow N $ be an immersed oriented hypersurface, $ e_{1}, \ldots e_{n},e_{n+1} $ be an orthonormal frame of $ N $ such that $ e_{1} \ldots e_{n} $ is an orthonormal frame of $ M $. Let ...
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548 views

Elliptic estimates on compact manifolds

Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it. For example, for $L$ a linear elliptic operator, (eg. ...
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47 views

Schoen Estimates (part 3)

I'm referring to the article 'Estimates for stable minimal surfaces in three dimensional manifolds' of Richard Schoen In the first paragraph of the proof of theorem 2 the author seems to assert that ...
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Riemannian metric for euclidean geometry

I am a beginner to Riemannian geometry. Following is my question. In the Euclidean space, say $\mathbb{R}^3$, let us consider a plane, for simplicity, say one passing through the origin, ...
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105 views

Regular compact domains of a Riemannian manifolds

In a Riemannian manifold $ M $ a regular compact domain $ D $ is a compact subset of $ M $ with non empty interior and such that for every $ p \in \partial D $ there exists $ \left(U,\varphi\right) $ ...
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64 views

Existence of vector fields

Does there exists two vector fields $X$ and $Y$ on $\mathbb R^2$ such that the following are satisfied? $X(0)= Y(0)= 0$, where $0\in \mathbb R^2$ and for others points $q\in \mathbb R^2$, we ...
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64 views

Schoen curvature estimates (part 2)

I'm referring to the article ''Estimates for stable minimal surfaces in three dimensional manifolds'' by Richard Schoen. I have a question about the proof of theorem 1. In the last step of the proof ...
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60 views

A question about an estimate

Let $ f:M \rightarrow N $ be a minimal immersion where $ M $ is a compact two dimensional manifold and $ N $ is a three dimensional manifold. Let $ |A|^2 $ be the square of its second fundamental ...
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355 views

Thoughts about sectional curvature

I'm currently trying to understand the sectional curvature of riemannian manifolds and I don't know if I'm thinking correctly. So, say we have a riemannian manifold $(M,g)$ with constant sectional ...
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259 views

why is the spectrum of the schrödinger operator discrete?

let (M,g) be a compact riemannian manifold. Then the spectrum of the Schrödinger opartor $H=-\Delta +V$ with bounded potential V acting on $L^2(M)$ consists of discrete Eigenvalues ...
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$|\Delta f|^2$ in local coordinates

The Laplace-Beltrami in local coordinates (for hypersurfaces in my case) is $$\Delta f = \frac{1}{\sqrt {|g|}} \partial_i \left(\sqrt{|g|} g^{ij} \partial_j f \right)$$ Is there a nice formula for its ...
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2answers
889 views

Divergence theorem on Hyperbolic space

Given a vector field, say $F$, defined on a manifold $U$, the divergence theorem states that: $$\int_U\nabla \cdot F dV=\int_{\partial U} F d \Sigma .$$ Well if the manifold is $\mathbb R ^n$ and $F$ ...
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Constant Riemannian Metric

Let $M=\mathbb R^n$ and define for each $x\in \mathbb R^n$, define $$\langle v,w\rangle_x= \langle v,w\rangle_0$$ where $v,w\in T_x\mathbb R^n\equiv \mathbb R^n\equiv T_0\mathbb R^n$. Hence we see ...
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302 views

Is there a non-variational derivation of Snell's law from Fermat's principle?

Every proof I've seen of Snell's law from Fermat's principle uses some sort of variational argument, mostly involving variational calculus. Niven's wonderful book, Maxima and Minima Without Calculus ...
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189 views

Manifold without conjugate points and positive curvature

I'm looking for an example of a complete riemannian manifold with sectional positive curvature and without conjugate points. I've tried the projective space, but the identfication used to construct it ...
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143 views

Riemannian metric. Help with notation.

I was just reading about the hyperbolic space (upper-half plane model) and i'm getting kind of confused about the notation for the Riemannian metric. The half-plane is defined as $$ H = \{(x,y) \in ...
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Riemannian Geometry book to complement General Relativity course?

What would be a good Riemannian Geometry (or Differential Geometry) book that would go well with a General Relativity class (offered by a physics department)? I'm in one right now, but I'd like a pure ...
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The standard connection in a regular surface is symmetric

The concrete setting: Let $M\subset \mathbb{R}^3$ be a regular surface. A vector field $X$ in $M$ is a differentiable function $X: M\to \mathbb{R}^3$ such that $X(p)\in T_pM$. Here we are taking the ...
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Are the Ricci and Scalar curvatures the only “interesting” tensors coming from the Riemannian curvature tensor?

In studying Riemannian geometry, one learns about the Riemann curvature tensor $$Rm(X,Y,Z,W) = \langle \nabla_X\nabla_YZ -\nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z, W\rangle$$ and its various symmetries. ...
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142 views

What are all isometry classes of the 2-sphere?

In topology, one learns how to classify the compact surfaces up to homeomorphism. And in fact, since "homeomorphic" and "diffeomorphic" coincide in dimension 2, we can classify the compact (smooth) ...
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What is the norm of the gradient of $f$ in normal coordinate?

Let $M$ be a Riemannian manifold and $f$ a smooth function on $M$. The Bochner formula proved in Schoen-Yau's book "lectures in Differential Geometry": (prop. 2.2) $$ \Delta |\nabla f|^2(p)=2\sum ...
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183 views

What does $ds^2$ mean and how does it specify a metric?

Let $H$ be the upper half-plane in $\mathbb{R}^2$. How does the following expression $$ds^2= \frac{dx^2+dy^2}{y^2}$$ specify a Riemannian metric on $H$? I don't understand what the expression means. ...
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25 views

Diagonable Kernels over a Riemannian Surface

This question is motivated by this paper. There, they develop a stippling method which requires a kernel to be diagonal. Meaning a symmetric bilinear function $K\colon \chi\times \chi\to \mathbb{R}$ ...
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184 views

Least distance on Riemannian Manifold

I've been doing some calculations of geodesics in different Riemannian Manifolds. More precisely I'm trying to compute, given two points on a Riemannian Manifold, the smallest distance between those ...
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166 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
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58 views

Bounding the injectivity radius from below.

Let $(M, g)$ be a finite-dimensional Riemannian manifold, and let $S \subseteq M$ be a compact set. I want to prove the following fact: There exists a number $\epsilon > 0$ such that the ...
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Critical paths for length cannot have kinks.

This problem is in Spivak's Differential Geometry (Ch.9 #37), and he gives a sketch of a proof which I have been unable to finish. So let's compute $\frac{dL(\overline{\alpha}(u))}{du}\mid_{u=0}$ ...
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809 views

How to understand the Second covariant derivative?

I am reading P. Li's lectures on Geometric analysis. On page 14, the author defines the second covariant derivative as follows: Let $f$ be a smooth function on $M$. $\omega_1, \cdots, \omega_n$ be a ...
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260 views

Connection between covariant derivative and basis vectors.

I read here, Curvilinear page 11, that $$\frac{\partial}{\partial x^i}e_j=\Gamma^k_{ij}e_k$$ where the $e_i$'s are basis vectors. There seems to be some connection, but when I calculate it, for ...
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What is the Weitzenböck formula for the $\bar\partial$-Laplacian?

It is well-known that the Weitzenböck formula for the real Laplacian is $$ \Delta |\nabla f|^2 =|\operatorname{Hess} f|^2 + \langle \nabla f, \nabla \Delta f\rangle + \operatorname{Ricci}(\nabla f, ...
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150 views

Length of closed geodesics in a compact and simply connected manifold X

I have a Riemannian manifold $(X,g)$ which is compact, simply connected and with sectional curvature upper bounded by $k>0$ everywhere. Let $p\in X$ be any point and $q\in Cut(p)$ the nearest cut ...
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328 views

control of the $C^{1}$ norm of a diffeomorphism

Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy: ...
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Prerequisites for studying Hodge theory and the Hodge conjecture

To what branch of mathematics does the Hodge conjecture belong? I'm aware that it's very advanced, but what kind of prerequisites would one need to understand those problems? Can you suggest some good ...
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140 views

Invariance of curvature under a conformal mapping

Let $\Omega_{1}, \Omega_{2} \subseteq \mathbb{C}$ be bounded domains. Let $\rho$ be a metric on $\Omega_2$ and $h: \Omega_1 \rightarrow \Omega_2$ a conformal mapping. Let $$h^*\rho(z) = ...
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1answer
239 views

showing zero curvature implies a line

How can I show that a given (not necessarily unit-speed) parametrization $\gamma(t)$ of a curve in $\mathbb{R}^3$ which exhibits zero curvature is a line ? What I know is that zero curvature means ...
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142 views

curvature of space curve

I am slightly confused by the following curve $\gamma(t) = (e^t,0,0)$ in $\mathbb{R}^3$. Its curvature, defined as $$ \kappa(t) = \frac{\|\dot \gamma(t) \times \ddot \gamma(t)\|}{\|\dot \gamma(t)\|^3} ...
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Computation of Laplace-Beltrami operator in a conformally equivalent metric

Could anyone tell me where I'm wrong with the following elementary calculation? Given a smooth Riemannian manifold $(M, g)$, I'd prove that if $\tilde{g}$ is conformally equivalent to $g$ (that is, ...
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99 views

Existence of Spin Group

"In mathematics the spin group Spin(n) 1[2] is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups As a Lie group Spin(n) therefore ...
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Is length adimensional when space is not flat?

Consider the two manifolds $\mathbb{R}^2$, equipped with the usual metric $g_{ij}=\delta_{ij}$, and $\mathbb{H}^2=\{(x, y)\,:\,y>0\}$, equipped with the hyperbolic metric $h_{ij}=\delta_{ij}/y^2$. ...
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538 views

good problem book in differential geometry

What are the books in Differential Geometry with good collection of problems. At present I am having John M.Lee's Riemannian Manifolds,Kobayashi Nomizu's Foundations of Differential Geometry. I ...
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155 views

Equivalence of intrinsic and extrinsic metrics of embedded manifolds.

Say a compact n-manifold $\mathcal{M}$ is embedded in $\mathbb{R}^m$, $m > n$. If $d_{\mathcal{M}}$ is the geodesic distance on $\mathcal{M}$, and $d$ the Euclidean distance in $\mathbb{R}^m$, ...
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Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?

I have previously asked about Weitzenböck identities and received some great answers on MathOverflow. One question which has arisen from the post is the following: Let $E$ be a hermitian ...
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68 views

Solving to get free falling coordinate as function of arbitrary coordinate

From weinberg's gravitation, EQ : $3.2.11$ $$\frac{\partial^2 \zeta^\alpha}{\partial x^\mu \partial x^\nu} = \Gamma^\lambda _{\mu \nu}\frac{\partial\zeta^\alpha}{\partial x ^\lambda}$$ The solution ...
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1answer
373 views

Dolbeault Cohomology is invariant under homeomorphisms

If $X$ and $Y$ are two complex manifolds, which are homeomorphic but not necessarily diffeomorphic, must their Dolbeault cohomology groups be isomorphic? Here the Dolbeault cohomology groups ...
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260 views

Covariant derivative of a pushforward

Suppose that $\Phi_t$ is a the global flow associated with a vector field $X$ on a Riemannian manifold $M$ and that $Y$ is any other vector field. Suppose furthermore that $X$ is a Killing vector ...