(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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surface in $R^3$ that has $ds^2 = du^2/v^2 + dv^2/v^2$

For a 2D surface, if we have the first fundamental form of $$ ds^2 = du^2/v^2 + dv^2/v^2$$, can we integrate it out to get the parameter form of the surface embedded in $R^3$? I tried something like ...
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29 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
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Two distinct geodesics joining two points on a compact manifold

This is a problem from the book Gallot, Hulin, Lafontaine: Riemannian geometry (3rd edition). Exercise 2.118: For a compact Riemannian manifold, let $p,q$ two points such that $d(p,q) = ...
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57 views

The computation of the Laplacian of the heat kernel on a Riemannian manifold

From John Roe, Elliptic Operators, topology and asymptotic methods , page 99 Let $M$ be a manifold of dimension $n$ with fixed point $q$. Let a geodesic local coordinate system $x^{i}$ originate from ...
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38 views

Minimizing geodesics don't have kinks

I'm working in a Riemannian manifold where all pairs of points are connected by a minimizing geodesic (i.e. a geodesic whose length equals the distance between the points). Here geodesics are ...
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30 views

Constant curvature geodesic circles on a surface with constant Gauss curvature

Referring to: Curvature of geodesic circles on surface with constant curvature, Is it possible to combine further the last two of the three equations in the link given above into a single ODE / PDE ...
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41 views

Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
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38 views

What is the adjoint of the connection operator on a Clifford bundle?

From Elliptic Operators, topology and asymptotic methods, John Roe, page 43-45. Let $M$ be a Riemannian manifold. Let $S$ be a Clifford bundle over $M$, such that each $S_{m}$ over $m\in M$ is a ...
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237 views

What meaning did Riemann assign to $dx$?

Detlef Laugwitz wrote a monumental biography of Riemann. The book was translated into English by Shenitzer. Laugwitz discusses Riemann's fundamental essay Uber die Hypothesen, welche der Geometrie ...
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23 views

An assumption used to derive the curvature tensor for Riemannian submersions

I was reading the literature about Riemannian submersions, and I came across the result showing the relation between the curvature tensor $\bar{R}$ in a manifold $M$ and the curvature tensor $R$ in a ...
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63 views

Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
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55 views

Is a Riemannian metric a $2$-form?

In Lee's Riemannian Manifolds; An introduction to Curvature, he defines a Riemannian metric as an element of $\Gamma(T^2_0M)$, a $(2,0)$-tensor. Is this the same thing as a $2$-form? Is there a ...
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manifolds with similar extrinsic and intrinsic distances ( locally)

Is there any specific name for those manifolds caracterized by having (locally) similar ( in some epsilon sense) extrinsic and intrinsic distances?
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37 views

Why is the matrix of a Riemannian metric positive definie?

Maybe I could post this as a linear algebra problem but I'll give some context. I know that if $(U, x_1, \ldots, x_n)$ is a local chart of a smooth manifold $M$ I can write locally a Riemannian ...
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18 views

Does the conformal class of a complex projective curve contain the Fubini-Study metric?

Let $X \subset \mathbb CP^2$ be a complex curve with metric $g$ induced by the Fubini-Study metric on $\mathbb CP^2$. Since in the case of two-dimensional real manifolds a complex structure is ...
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27 views

Example of complete not-connected riemannian manifold

Are there examples of complete Riemannian manifolds which are not connected ? This question follows my previous question. The more I think about it and the less I'm convinced it exists.
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63 views

Riemannian metric and geodesic completeness

For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$. I am not sure about the ...
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37 views

When do isometries commute with the compatible derivative operator on a semi-Riemannian manifold?

Let $M$ and $\tilde{M}$ be smooth manifolds, each with a metric $g_{ab}$ and $\tilde{g}_{ab}$, assumed here to be smooth symmetric invertible tensor fields, which are non-degenerate but not ...
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60 views

Ricci Tensor, Curvature and Scalar Curvature computation from definition

I am studying a little of Riemannian geometry and I am having some problem in making the connection between two expressions of Ricci tensor, curvature and scalar curvature. Well, in the book that I am ...
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29 views

Question on Torsion free condition for Levi-Civita connection

I was watching a video on Riemannian Geometry. The lecturer mentions that given the defining condition for a connection on a Riemannian manifold $M$ i.e. : $$\nabla_X(Y) : \chi(M) \times \chi(M) \to ...
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44 views

Parallel Transport of Geodesic Velocity Vectors

Given a Riemannian manifold $M$ with Riemannian metric $g_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ and distance $d:M\times M\rightarrow\mathbb{R}$ determined by length of minimizing geodesics, ...
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236 views

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from ...
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1answer
41 views

Levi Civita connection along principal curvature directions

Let $(M,g)$ be a surface that can be immersed into $\mathbb{R}^3$. Denote by $\nabla$ the associated Levi Civita connection. Further, let $X_1,X_2$ be the directions of principal curvature which are ...
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1answer
30 views

Difference betwee parameterization and embedding of manifolds

What is the difference between embedding and parameterization? Why, for example, we say Gauss parameterization of a convex hypersurfaces, and we don't call it an embedding?
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Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
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Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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44 views

extension of a local orthonormal frame on a hypersurface

Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along ...
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75 views

relative sign in Hodge star of tensor product

Let $V$ be a vector space of arbitrary (finite) dimension and let $(V, \langle \ ,\ \rangle, I) = (W_1, \langle\ ,\ \rangle_1, I_1) \oplus (W_2, \langle\ ,\ \rangle_2, I_2)$ be a direct sum ...
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1answer
68 views

What is the meaning of the symbol $\nabla^k$?

In T.Aubin's book, a course in differential geometry, he write the formula $\Delta f=-\nabla^k\nabla_kf$ on a Riemannian manifold, but he never define the symbol $\nabla^k$. It seems that the notation ...
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1answer
41 views

Riemannian Metric Notation

I am just being introduced to Riemannian metrics, and I am having a bit of confusion on the notation. When reading, I've encountered some different notation in different sources, so I want to make ...
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68 views

Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
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How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$ X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3 $$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$ x=r \cos(t) \\ y=r \sin(t) $$ ...
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1answer
31 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
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1answer
115 views

Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
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1answer
38 views

Does the possibility of linear coordinate changes imply that the manifold is Euclidean?

Question. Let $M$ be a smooth manifold admitting an atlas $\mathcal{A}$ (i.e. a collection of coordinate charts that covers the whole manifold) with the following property: for every pair ...
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51 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
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1answer
25 views

What is the exterior normal to the boundary of a Riemannian manifold?

Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $p \in \partial M$ and in local coordinates $(x_1,\ldots,x_n)$ near $p = (0,\ldots,0)$ the manifold $M$ is given by $\{x_n ...
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Question about a particular estimate in Riemannian geometry.

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...
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1answer
35 views

Constant in Sobolev-Poincare inequality on compact manifold $M$; how does it depend on $M$?

Let $M$ be a smooth compact Riemannian manifold of dimension $n$. Let $p$ and $q$ be related by $\frac 1p = \frac 1q - \frac 1n$. There is a constant $C$ such that for all $u \in W^{1,q}(M)$ ...
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49 views

Reference for construction on Riemannian Manifolds

I would like to know a book or an article where the connected sum of Riemannian manifolds is explained with some detail. I roughly know how to do the construction but I want to be able to check a more ...
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1answer
61 views

Riemannian Submerssion

I am reading John Lee's Riemannian Geometry Chapter 3, and I want to do some exercises. I think that I need some hints to solve the following: (Problem 3-8 of that book) Suppose $M$ and $N$ are ...
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1answer
68 views

Why doesn't a metric give an isomorphism $TX \cong T^*X$?

Any smooth manifold $X$ admits a Riemannian metric $g$, and we have a map $$ TX \to T^*X, \qquad (x, v) \mapsto (x, g(v,-)) $$ which is smooth if $g$ is. Why isn't this an isomorphism of vector ...
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Product neighborhood theorem with boundary

The Product Neighborhood Theorem states that if $N\subseteq M$ is a smooth compact submanifold without boundary of codimension $n$ and there is a trivialization of the normal bundle (wrt. some smooth ...
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Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
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Perturbation of the boundary of a strictly stable minimal surface

Let $\Sigma \subseteq \mathbb{R}^3$ be a minimal surface with boundary $\Gamma$. Now let us assume that $\Sigma$ is strictly stable, that is, $\lambda_1(\Sigma,L) >0$, where $L$ is the stability ...
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Ellipticity of Ricci tensor, does it depend on coordinates?

Well, I am afraid this is a silly question because I know the answer must be 'yes, it does'. But I don't see why. I put the problem in context. The ricci tensor can be regarded as a differential ...
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1answer
43 views

What are the laplacian operators for the three two dimensional metrics of one variable dependence?

What are the laplacian operators from the three following two dimensional metrics of one variable dependence : \begin{align} (A) && d\mathcal{l}^2 = e^{2w(x_2)} \left( dx_1^2 + dx_2^2 \right) ...
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Known methods to detect coordinate singularities?

MOTIVATION In Riemannian geometry, when one writes the metric tensor in a particular coordinate system, certain 'fake' singularities might appear that have little to do with the geometry near them. ...
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Reference frame rotation depending on metric tensor $g_{\mu\nu}$

My simple question is: The infinitesimal line element is $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$$ where $g_{\mu\nu}$ is the metric tensor of the space. Is it possible from the simple knowledge of ...