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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Ellipticity of an operator in Gunther's proof of the isometric embedding

In Deane Yang's notes about Gunther's proof of the celebrated isometric embedding theorem, at the end it is stated that $v$ inherits the regularity of $h$ because the operator $I-Q_0(v,\cdot)$ is ...
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Proof of the fundamental inequality of the index form

I am looking for a proof of the fundamental inequality of the index form, which I have seen as references or statements in a lot of sources, but without a proof. This is the statement: Let $M$ be a ...
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63 views

saddle point geometry of $f(x,y)$ when partial derivatives are of same sign.

The sign of Hessian matrix and sign of any partial derivative of function $f(x,y)$ gives the informations about maxima, minima, saddle points of function and sometimes perplexed informations when it ...
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153 views

Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...
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Newton iteration on Riemannian manifolds

Suppose $f:M \to N$ is a smooth map between complete Riemannian manifolds of the same dimension. Suppose $Df(m_0)$ is invertible, and $n$ is a point close to $f(m_0)$. Can we perform Newton iteration ...
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ricci tensor of 2-sphere $S^2$

Hi could someone show me explicitly how to compute the ricci tensor $g_{ij}$?
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Mean Curvature flow: Evolution equation of any invariant symmetric homogeneous polynomial with input the Weingarten map.

I have the following evolution equations realted to mean curavture flow, with the induced metric $g=\{g_{ij}\}$, measure $d\mu$ and second fundamental form $A=\{h_{ij}\}$: 1)$\frac{\partial}{\partial ...
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47 views

An identity involving a Killing field

Does anyone know how to prove the following identity. We assume that $\Omega$ is a Killing field and $U, V$ are vector fields. Then $[\Omega ,\nabla _UV]-\nabla _U([\Omega, V])=\nabla _{[\Omega ...
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121 views

Homogeneous Isotropic Riemannian Manifolds

In John Lee's book Riemannian Manifolds on page 33, Lee writes "Clearly a homogeneous Riemannian manifold that is isotropic at one point is isotropic at every point". It seems that he means that he ...
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35 views

Pulling back a connection to a curve

What does it mean to pull a connection back to a curve? For example, if I take the connection $\nabla s = ds$ on the trivial bundle $\mathbb R^2 \times \mathbb R^2$ over $\mathbb R^2$, and the curve ...
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66 views

Curvatures in differential geometry-interpretation

The are various notions of curvatures in differential geometry: soft such as full curvature tensor for a given connection (which is tensor of type $(1,3)$), Ricci curvature tensor (type $(0,2)$ ...
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Metric Isometry is always smooth?

Let $M$ be a smooth manifold. Let $d$ be any metric on $M$ which induces the topology on $M$. Let $f:(M,d) \rightarrow (M,d) $ be an isometry (in the sense of metric spaces). Is it true that $f$ must ...
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Error when computing geodesics in hyperbolic half plane

It is known that the geodesic equations for the upper half plane equipped with the hyperbolic metric are $$x''=\frac{2x'y'}{y},$$ $$y''=\frac{(y')^2 -(x')^2}{y}.$$ It is also well known that the ...
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64 views

If $\nabla_1$ and $\nabla_2$ are Levi-Civita connections for a metric on the smooth sphere, then their curvature tensor would recover the radius…?

I am a little confused by an idea suggested to me: putting a connection on a sphere doesn't specify a metric geometry - it remembers notions like straightness of paths, but not length of paths. Let ...
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45 views

Holonomy computation in a sphere

Let $S^1$ be the unit sphere in $\mathbb R^3$, and let $$C=\{(r\cos t, r\sin t, h)\colon t\in \mathbb R\}$$ with $r^2+h^2=1$ be a circle in $S^2$. I want to compute the holonomy around this circle. ...
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48 views

Diffeomorphism that pulls back the curvature tensor is an isometry?

I heard this statement somewhere. Can anyone provide a reference (or explanation of why this is true)? (I have also heard that the metric can be expanded as a power series in terms of the curvature ...
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23 views

Applications of normal coordinates?

I am looking for applications of normal coordinates (from the exponential map) on a Riemannian manifold, since I am trying to familiarize myself with this notion. Presently the only one that I know ...
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51 views

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ bilinear … so is it tensor like?

The function $m: C^{\infty}(M) \times C^{\infty}(M) \to C^{\infty}(M)$ $m(f,g) = fg$ is $C^{\infty}$ linear in both components... so is it a tensor of some kind? I know (I think) it is not a ...
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Space of $G$-connections; respecting a spin structure

If I want to have a space of $G$-connections on a Riemann surface, I can take the fundamental group on the surface, represent its generators on $G$ and take (up to conjugation) those representation ...
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83 views

Computing curvature of hyperbolic space

Consider the unit ball in $\mathbb R^2$ endowed with the Poincaré metric: $$ds^2=\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}.$$ I want to compute the Gaussian curvature and find that it is $-1$. Given that ...
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74 views

Chern-Gauss-Bonnet theorem for even-dimensional manifolds with boundary

On the wikipedia page for the Chern-Gauss-Bonnet theorem it states that there is a generalization of the theorem for even-dimensional manifolds with boundary, but does not provide the relevant theorem ...
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Constructing a metric for the tautological line bundle of $\mathbb C P^2$

I'm doing some independent reading in differential geometry, and the following is my attempt to work out the details of the construction of the tautological bundle on $\mathbb CP^2$ and the induced ...
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66 views

Is Hilbert's theorem generalizable to $H^3$ immersion in $\mathbb{R}^4$?

The Hilbert theorem in differential geometry concerns the immersion of the hyperbolic plane in $\mathbb{R}^3$. Is it valid for $H^3$ in $\mathbb{R}^4$?, and for all $H^n$ in $\mathbb{R}^{n+1}$?
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68 views

Complete Riemannian metric on ${\mathbb R}^2\setminus\{0\}$.

It seems to me that the Riemannian metric $g_{ij}=\delta_{ij}/|x|^2$ on the punctured plane is complete, but I don't find a proof not involving explicit computations of the geodesic equation. Does ...
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Taylor series representation for a Riemannian hypersurface

This is Exercise 8.5 in Lee's Riemannian Manifolds: An Introduction to Curvature. Suppose $M\subset \mathbb R^{n+1}$ is a hypersurface with the induced metric. Let $p\in M$, and let ...
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31 views

a problem in gauss lemma

I was reading the Gauss Lemma from the do carmos Rienmannian geometry book which says that Let $p \in M$ and let $v \in T_pM$ such that $\exp _p v$ is defined Let $w\in T_pM$ is identified with ...
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41 views

Equidistant points to a hyperbolic line

consider the Poincare upper half-plane model of hyperbolic plane $\mathbb{H}^2$ and a hyperbolic line $\ell\subset \mathbb{H}^2$ (or geodesic if you want). I would like to visualize the set of points ...
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looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
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22 views

Geodesic connectivity implies geodesic convexity?

Assume a subset $C$ of a Riemannian manifold $M$ is "geodesically-connected", that is: given any two points in $C$, there is a geodesic contained within $C$ that joins those two points. Is it true ...
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146 views

Distance of two hyperbolic lines

Consider the upper-half plane model of the hyperbolic plane $\mathbb {H}^2.$ Now consider two lines in it given as $\ell_1:=\lbrace { (x, y)\in \mathbb {H}^2 \vert x^2 +y^2=r^2\rbrace}, ...
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Countable intersection of Cut Locuses is always empty?

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then does there always exist a countable collection of points $\{p_n\}_{m \in \mathbb{N}}$ such that: \begin{equation} ...
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If $N$ is the boundary of Riemannian $M$, can I compute $i^{*}(* (\alpha \wedge \beta))$?

There wasn't enough room in the title to explain completely: $M$ is an oriented Riemannian manifold with boundary $N$. $\alpha$ and $\beta$ are differential forms on $M$, $*$ denotes the Hodge star, ...
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48 views

Classical differential operators with complex functions on Riemannian manifolds

I am having some trouble understanding how to use the classical operators ($\nabla, \operatorname{div}, \Delta$) with complex functions on a Riemannian manifold $(M, g)$. Consider the formula ...
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60 views

2-dimensional Riemann Manifold

I am looking for a proof of the theorem that states that any 2-dimensional Riemann Manifold is conformally flat in the case of a metric of signature 0, following through with Problem 6.30 in the text ...
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The Riemannian Distance function does not change if we use smooth paths?

The Riemann distance function $d(p,q)$ is usually defined as the infimum of the lengths of all piecewise smooth paths between $p$ and $q$. Does it change if we take the infimum only over smooth ...
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Computation in Wikipedia's article “Riemann Curvature Tensor”

This Wikipedia article explains how the Riemann curvature tensor is a measure of the failure for a tangent vector to parallel translate back to itself along an infinitesimally small loop. The article ...
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Triangle equality in a Riemannian manifold implies “geodesic colinearity”?

Let $(M,g)$ be a non-complete Riemannian manifold. Assume $p,q,r\in M$ satsify: $d(p,q)=d(p,r)+d(r,q)$, where $d$ is the Riemann distance function induced by the metric $g$. I am trying to find the ...
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50 views

Diffeomorphism between $\Bbb{R}^{4}$ and the cube

I'm looking for an explicit diffeomorphism between the four-dimensional euclidean space $\Bbb{R}^{4}$ and the four-dimensional open cube. I wonder whether there is a simple looking map, with simple ...
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1answer
31 views

Intersection of Cut Locuses

If $C_p(M)$ is the cut locus of some $p\in M$ in some Riemannian Manifold $M$, then when is: \begin{equation} \bigcap_{p\in M} C_p(M)=\emptyset\text{ ?} \end{equation}
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138 views

Riemannian metrics and how spaces look

I thought I had a fairly good understanding of Riemannian metrics until I came across this exercise in Petersen's book. Construct paper models of the Riemannian manifolds ($\mathbb{R}^2, dt^2 + ...
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35 views

geodesic of Stiefel manifold

Define a metric on Stiefel manifold $V_{n,p}$ as $$\left<\Delta_1,\Delta_2\right>=\text{tr}\Delta_1^T\left(I-\frac{1}{2}YY^T\right)\Delta_2$$ $\forall \Delta_1,\Delta_2\in T_YV_{n,p}$ how to ...
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44 views

N-polygons in hyperbolic geometry

Let $N$ be an integer and we have two $N$-polygon $A_{1}A_{2}\ldots A_{N}$ and $A'_{1}A'_{2}\ldots A'_{N}$ such that the length of geodesic $A_{i}A_{i+1}$ is equal to the length of geodesic ...
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Reference needed for a short time existence result of quasilinear PDE on a compact manifold (relating to Ricci flow).

I'm currently in the proces of learning and writing a bit about the Ricci flow. In particular I'm studying the case of compact 2d Riemannian manifolds. Mostly I'm making good progress but I do miss ...
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Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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50 views

Which Riemannian Manifolds are scale-free?

Let us call a Riemannian manifold $(M,g)$ scale-free if for any real positive scalar $\lambda$, $(M,g),(M,\lambda g)$ are isometric. $\mathbb{R}^n$ with the standard metric $g$ is scale-free. (Via ...
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Lie bracket is part of the intrinsic “geometry”? But I have seen it defined without a metric…?

I have seen two definitions of the Lie Bracket for a Riemannian manifold $(M,g)$. One is this : $[X,Y] = D_X Y - D_Y X$, where $D$ stands for covariant differentiation. When written out, this seems ...
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59 views

Curve with with curvature $k(s)\ge 1$ everywhere has diameter $\le 2$

Let $\alpha(s)$ be a simple closed plane curve. Define the diameter $d_\alpha$ of $\alpha(s)$ to be $$d_\alpha = \sup_{t,s\in\mathbb R} \| \alpha(s) - \alpha(t) \|.$$ Assume the curvature ...
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228 views

Local isometries preserve geodesics?

Question: It is well known that if $\varphi:M\to \tilde{M}$ is an isometry between Riemannian manifolds, then $\varphi$ maps geodesics of $M$ to geodesics of $\tilde{M}$. I am wondering if it is ...
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53 views

Question of well-definedness of the Levi-Civita connection?

On page $55$ of Do Carmo's Riemannian geometry, he proves that there is a unique symmetric affine connection compatible with a given metric on a manifold M. He defines it by a formula $\langle Z, ...
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51 views

Riemannian Geometry notational tricks or alternatives

I am interested in learning tricks that people have developed to speed up / clean up calculations in Riemannian Geometry. I am hopeful about this question because there is often a lot of symmetry in ...