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Questions tagged [riemannian-geometry]

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 if your question involves Riemannian manifolds or objects generally associated with them, such as Levi-Civita connections.

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How to derive the Euler Lagrange equation for geodesics?

In my book, it says a geodesic is associated to the functional $\int_0^l |\gamma'|^2$ , with a metric g. It then jumps to $\ddot{\gamma}^k + \Gamma^k_{ij}\dot{\gamma}^i\dot{\gamma}^j = 0$ where $\...
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Ricci equation is trivial in codimension $1$

Let $f:M\to\overline{M}$ an isometric immersion and assume $\dim(M)=\dim(\overline{M})-1$. I'm asked to show that the Ricci equation offers no information. I guess what I have to show is that the ...
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Why is the matrix representation of an almost complex structure like this?

Let $M$ be a Riemann surface, $J$ be an almost complex structure (i.e. a 1-1 tensor such that $J^2=-I$ and for any $x \in M,v,w \in T_xM, \{v,Jv\}$ is oriented). Consider a conformal coordinate at a ...
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Two definitions of a connection

For me a connection $\nabla$ on a vectorbundle $E$ over a smooth manifold $M$ is a $\mathbb{R}$-bilinear map $\Gamma(TM)\times\Gamma(E)\rightarrow\Gamma(E)$ which is tensorial in the first slot and ...
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Soft question: why define smooth manifolds intrinsically?

A possibly naive question, but one I've been grappling with since starting Riemannian geometry. I have done some searching and I cannot find this question asked, yet it feels like a very natural one ...
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Biconformal space and curvature

I've found very few contributions about the so called Biconformal Space, "a curved phase space". I was sure that in general phase spaces are cotangent bundles naturally equipped with a symplectic 2-...
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Gauss formula along a curve

On page 138 of "Riemannian Manifold: An introduction to curvature" I can read the following statement: Let $S$ be a Riemannian submanifold of $M􏰄$, and $\gamma$ a curve in S. For any vector field $...
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Principal Symbol for Ricci-DeTurck Flow

I am following some lecture notes on Ricci flow and linearizing an operator to obtain its principal symbol. We have $T \in \: \Gamma(Sym^2 T^{*}M)$ smooth, fixed and positive definite and then ...
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Induced Maps from Sections

I am reading a book on Riemannian geometry and it says 'let $T \in \:\Gamma(Sym^2 T^{*}M)$ be fixed, smooth, and positive definite. We also denote by $T$ the invertible map which $T$ induces, given a ...
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How to calculate the derivative of a scalar curvature for a Ricci flow?

Now we have a homogeneous ricci flow, which means the initial data is homogeneous and then for each $t$, $g(t)$ is still homogeneous. My question is what is $$\text{scal}(g(t))'$$ In a paper(sec 3,...
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What is the time doubling property of Riemannian curvature tensor in a homogeneous ricci flow?

As in the title, I read this in the paper Optimal Curvature Estimates for Homogeneous Ricci Flows about Homogeneous ricci flows. In section 3, this "time doubling property" appears and is used in the ...
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Tangent vector field to a smooth curve over a smooth manifold

I am teaching myself some elementary differential geometry and am stuck on the concept of the tangent vector field of a smooth curve. I have searched the web for an hour or so but cannot find anything ...
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Definition of closed surface/manifold

This question might appear silly, I was reading on wikipedia that a closed surface (or manifold in general) is a surface without a boundary, I'd like to elaborate a bit on such definition. Assuming ...
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Is this a correct definition of a Riemannian metric tensor?

I am a total beginner in Riemannian geometry but I'm trying to teach myself the basics. So the following could contain many horrible mistakes. Suppose I describe the x,y-plane by curvilinear ...
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2answers
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Computing volume of Riemannian manifolds

I have a question- It is given that $f: M \mapsto N$ is an $n$- sheeted covering map and a local isometry then I have to show that volume$(M) = n$ volume$(N)$, where $M$ and $N$ are Riemannian ...
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Inner product on the Hyperbolic half plane

I think this is the dual question to my previous question. From Gudmundsoon notes, page 60: We can model the hyperbolic space $\mathbb H^m$ as the super half plane space $\mathbb R^+ \times \mathbb ...
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Isometries of a surface with metric of curvature $-1$

Let $\mathbb{H}^2$ be the hyperbolic space in the model $$\mathbb{H}^2=(\mathbb{R}\times\mathbb{R}_+,g=\frac{1}{y^2}(dx^2+dy^2)).$$ It is known that the Mobius transformations, with $ad-bc=1$, are ...
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An Exercise about Riemann Geometry

Let G be a compact Lie group with a bi-invariant metric. (a)Let p be a point, and let q be conjugate to p along a geodesic . Show that the dimension of the space of Jacobi fields along vanishing ...
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Definition of the surface measure in some Book

I am studying PDEs and in some place I found an integral integrated by the surface measure on M=($C^k$ - Hypersurface of $R^n$). 1)Is there any reference to see how are defined this Measure on M? 2)...
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how we can add a 2-tensor to a 1-form?

In the definition of a Randere norm, we add a Riemannian metric $\alpha$ to a 1-from $\beta$. Indeed, $F(y)=\sqrt{a_{ij}(x)y^iy^j}+b_i(x)y^i$ in ehich $\alpha(y)=\sqrt{a_{ij}(x)y^iy^j}$ is a ...
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How would the chart around a point vary be different to the tangent space at the same point?

If I'm not mistaken, the tangent space and the space that a chart maps to are both Euclidean. So if I take some points from a neighbourhood around a point on the manifold, and map them to Eulclidean ...
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Change in volume form under exponential map

We now that exponential map preserves length of radial tangent vectors as well as orthogonality of radial and spherical tangent vectors (guass’s lemma). How does volume form change under the ...
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Can we 'build' spinor structure not only from a Riemann Manifold but 'extract it' also from another algebraic structures?

I want to understand what type of structures are Spin Structure: are a monoids, ringoids, groups? Can we build spinor structure find also from another structures not 'extract it' only from a Riemann ...
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Isometry group of a lorentzian metric which preserves a Riemannian metric

I'm wondering if it is possible for the isometry group of a Lorentzian metric to preserve a Riemannian metric. So if $(M,g)$ is a Lorentz manifold and $Iso(M,g)$ its isometry group, I ask myself if ...
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Non-conformal metrics on vector bundles where $\nabla g=\omega(\cdot) g$

Let $E$ be a smooth vector bundle over a manifold $M$ ($\dim M \ge 2$), equipped with a metric $g$ and a connection $\nabla$, such that $\nabla_X g=\omega (X) g$ for every vector field $X$ on $M$. ($\...
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1answer
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Does the concept of “adjoint map” determine the metric up to scaling?

Let $V$ be a real finite-dimensional vector space, and $g$ and inner product on $V$. $g$ induces a concept of "adjoint map" , i.e. a linear map $\text{Hom}(V,V) \to \text{Hom}(V,V)$ given by $S \to S^...
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First Area Variation

Could someone help me to solve this limit? Let $X:\Omega \subset \mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$ imersion and ${\LARGE \chi }_{t}:\Omega \subset \mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$, ...
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Question about circles in hyperbolic space

I'm trying to prove the following: In the half-plane model of $\mathbb H^3$ let $L$ be the geodesic going from the origin of $\Bbb R^3$ to $\infty$. So $L$ is a straight half-line perpendicular to $\...
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Example on Local Systems: Flat Connections

I am reading about the local system, and a local system $\mathcal{L}$ on $X$ with values in $\mathcal{C}$ is defined as a functor $$ \mathcal{L}:\Pi(X)\to \mathcal{C} $$ where $\Pi(X)$ is the ...
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An example of smooth but not Riemannian [duplicate]

I've been trying to understand the difference between the notion of smooth (which I understand well) and Riemannian (which I am newly acquainted with). The definition in the tag for 'riemannian-...
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curvature tensor of involutive distributions

The Riemann curvature tensor is defined $$R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y \nabla_XZ - \nabla_{[X,Y]}Z.$$ Is it possible to simplify above expression under assumption that $X,Y\in D$, where $D$ ...
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Covariant derivative induced by Levi-Civita connection and compatibility with Lie brackets

Let $M$ be a Riemannian manifold with Levi-Civita connection $ \nabla$. Let $S$ be a differentiable manifold and $ \varphi : S \to M $ be a $C^{\infty}$ immersion. Let $$ D : TS \times \{ \text{...
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Intuition behind Riemannian-metric

I apologise in advance if something like this has been asked already and I will delete this question immediately if an already answered question of this sort clears my doubt, which is- What is a ...
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Matrix Multiplication on Riemannian Manifolds

I am having a hard time understanding the concept of matrix (and / or vector) multiplication on a Riemannian Manifold $(M, g)$. On $\mathbb R^n $ we can multiply a matrix for a vector in the usual ...
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Riemann hyper-spheres for hyper-complex numbers?

Today I learned that the Riemann sphere can map the extended complex plane to the surface of a sphere. It's straight-forward to show that an analogous mapping can be found for a line and a circle. I'...
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Summary and understanding of non-euclidean Geometry

I'm trying to understand the 'paradigm shift' from Euclidean to non-euclidean geometry. Though I can understand simple models like why the angle in a triangle on a sphere would not add to 180 degrees ...
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How to read the expression of an affine connection: $\nabla_X Y$?

I am studying Riemannian Geometry from the textbook Riemannian Geometry by do Carmo (English edition). In section 2 of chapter 2, page 50, he defines an affine connection as follows: 2.1 Definition....
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Is convexity radius at a point of a complete Riemannian manifold always smaller than injectivity radius at that point?

Is the convexity radius $\mathop{\mathrm{conv.rad}}(p)$ at a point $p\in M$ of a complete Riemannian manifold $M$ always smaller than the injectivity radius $\mathop{\mathrm{inj}}(p)$ at that point? ...
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1answer
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Uniform distribution on Stiefel

I want to implement the method of sampling (uniformly) points on Stiefel manifold but I'm failing to find any kind of research/article/work that can give some info about the methods and techniques of ...
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Does parallel transport change the subspace?

Let $M$ be a Riemannian manifold and $N$ be an immersed submanifold. Take $\gamma$ a geodesic starting at and perpendicularly to $N$. Let $X(t)$ be a vector field along $\gamma$ such that $X(0) \in T_{...
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1answer
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Characterizations of Riemannian Volume Form

I'm trying to understand how some characterizations of the Riemannian volume form $dV$ are equivalent on an oriented Riemannian manifold of dimension $n$. I'm a bit new to Riemannian geometry (and ...
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Taylor Expansion in Normal Coordinates

Let $\phi: (M,g)\hookrightarrow (N,\tilde{g})$ be an isometric embedding of a Riemannian manifold $M$ of dimension $m$ into a Riemannian manifold $N$ of dimension $n$. I am interested in trying to do ...
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Adapted connection on foliation

Let $(M,F)$ be a manifold with foliation, fixing a bundle-like metric $g$ we identifying $Q=TM/F\cong F^\perp$, we define the connection on $Q$, for any $s\in\Gamma(Q)$ $$\nabla_Xs=\begin{cases} ([X,...
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Covariant derivative of a symmetric tensor

Assume that a symmetric $(0,2)$ satisfies $$\nabla_iT_{jk}+\nabla_jT_{ik}+\nabla_kT_{ji}=0$$ where $T=T_i^i$ is constant and $\nabla_jT_{ik}\ne 0$. What are the values of the constants $a,b,c$ such ...
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Reverse Toponogov triangle comparison: reference or proof (Riemannian geometry)

The Toponogov comparison theorem allows to bound the distance between two points in a hinge from above using the minimum curvature of the Riemannian manifold. This Wikipedia page states that it can ...
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A question about the relation between the exterior derivative of $1$-forms and the metric

Let $\theta_X$ be a $1$-form. Petersen's "Riemannian Geometry" says the following on pg 24: $d\theta_X(\partial_k,\partial_l)=\partial_kg(X,\partial_l)-\partial_lg(X,\partial_k)-g(X,[\partial_k,\...
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Existence of a parallel vector field implies a splitting of the metric

Where can I find a proof of the following claim: Existence of a parallel vector field on a Riemannian manifold implies that the metric splits locally as a product of a one-dimensional manifold and $n-...
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A question about a claim from “Riemannian Geometry” by Petersen

Petersen says the following on pg 13 of "Riemannian Geometry" by Petersen: If $\frac{\psi^2-1}{t^2}$ is a smooth function of $t$ at $t=0$, then $\psi^{(1)}=\psi^{(3)}=\dots=\psi^{(odd)}=0$ I don't ...
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103 views

What is a differetial structure, exactly?

A structure in general is a set and some operations on that set or ordersrelations of some kind. In algebra and topology this is rather clear, but in differential geometry one often consider "...
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Extension of Du-Bois-Raymond lemma to Vector Fields on a Riemannian Manifold

I am trying to show the following extension of the Du Bois Raymond lemma: Let $M$ be a smooth Riemannian Manifold and $\omega: [0,1] \rightarrow M$ be a $W^{1,2}$ curve on M. Consider a tangential ...