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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Lipschitz Continuity for a function on stable minimal hypersurface immersed in $\mathbb{R}^n$

I'm going through a proof of Schoen-Simon-Yau's $L^p$ bound on the norm squared of the 2nd f.f., $|A|^2$, for stable (orientable) minimal hypersurfaces in $\Sigma \subset \mathbb{R}^n$, from Minicozzi ...
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Time dependent one-form change under Ricci flow

So this is part of a proof in Peter Topping's text, Lectures on Ricci Flow that I don't understand. Let $\delta A=-\text{tr}_{12}\nabla A$ and $\omega$ be a time dependent one-form. Also we use the ...
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17 views

A geodesic which stop minimizing must self intersect?

Let $M$ be a complete* Riemannian manifold. Let $\gamma$ be a geodesic which stops minimizing at some point. Is it true that $\gamma$ must be periodic or self-intersect? (in a transversal way) I ...
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Trace of hessian

So this is a result used in Peter Topping's,Lectures on Ricci Flow. What is a quick of showing $$\text{tr}\nabla_{X,\cdot}^2h(\cdot,W)=-(\nabla\delta h)(X,W)$$ where $\delta A=-\text{tr}_{12}\nabla ...
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26 views

Bounded set in Riemannian manifold is relatively compact

Suppose $\{x_n\}_{n=1}^{\infty}$ is a bounded sequence in a finite dimensional Riemannian manifold. Can I say that this set is relatively compact? In $R^n$, this is true by Bolzano-Weierstrass ...
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Conformal mappings between open disk and half space

In which book / handout can I find an explicit description of a conformal mapping between the open ball and the upper half space in $n$ dimensions?
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95 views

The second integral of the Killing form

Let $G$ be a lie group. Assume that $B$ is the Killing form of its Lie algebra $T_{e}G$. So $B$ is counted as a symmetric $2$-form on $G$ by translation. Is there a smooth function $f$ on $G$ ...
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Are there surfaces that have (except for cusps or borders) a constant positive gaussian curvature but that do not have not a constant mean curvature?

I was puzzeling with the pseudosphere , a surface that except for a cusp has a constant negative Gaussian curvature, but has not everywhere the same mean curvature. ...
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62 views

Levi civita notation

I'm having troubles with the Levi-Civita symbol. I understand what the normal epsilon-tensor means and how it works. But how do I interpret this: ...
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51 views

Example of Skew-Symmetry of Connection Forms

As is commonly known, the connection 1-forms of a Riemannian manifold are skew-symmetric: $\omega^i_j=-\omega^j_i$. Until now, I have not actually thought to hard on this, but I think I've hit a snag. ...
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Manifolds with geodesics which minimize length globally

I am interested in complete Riemannian manifolds whose geodesics minimize length globally. Such manifolds must be non-compact (otherwise there is always a self-intersecting geodesic) However, I ...
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Hermitian metric from Killing form

Let $G$ be a semisimple Lie group. Its Killing form is a nondegenerate inner product on the tangent space to $G$ at the identity, and this form can be naturally extended to a metric on the whole of ...
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How to show $(\nabla df)(X,Y)=\nabla^2_{X,Y}f$? [duplicate]

Why $(\nabla df)(X,Y)=\nabla^2_{X,Y}f$ ? I only know $df(Y)=Y(f)$, I always see $df$ as an element of $T^*M$,and I think $df=f_idx^i,f_i=df(\partial_i)$. Besides, whether $(\nabla ...
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61 views

Why define $(\nabla^2F)(X,Y)=\nabla_X(\nabla_YF)-(\nabla_{\nabla_XY}F)$?

Why define $$(\nabla^2F)(X,Y)=\nabla_X(\nabla_YF)-(\nabla_{\nabla_XY}F)?$$ I can't find the motivation of this definition .I don't know the purpose of defining so. The more details the better, I am ...
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25 views

The Levi-Civita connection on $S^3$ and $SU(2)$

The fundamental theorem of Riemannian geometry implies that there is a unique symmetric (i.e., $\Gamma^{a}_{bc}=\Gamma^{a}_{cb}$, using a coordinate basis) connection on the three-sphere, $S^3$ which ...
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27 views

Covariant derivative of 1-form

I use the 3 of Proposition 1.27 to compute equality above red line. $\nabla_X(dx^j\otimes\partial_i)= \nabla_X(dx^j)\otimes\partial_i+ dx^j\otimes\nabla_X(\partial_i)$. Then, how to get the red line ...
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36 views

Can a hyperbolic surface be isometrically embedded into $\mathbb R^4$?

Can a complete hyperbolic surface be isometrically embedded into flat $\mathbb R^4$?
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How to understand $g^{ij}\omega_j=\omega^i$?

What is $\omega_j$ ? I remember the index of covariant tensor is in the upper right corner ,like this $\omega^j$. And $\omega$ always represent covariant tensor.So, I'm fuzzy with $\omega_j$. In ...
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4answers
49 views

What is mean of $\dot{\gamma}^j(t)$?

In picture below ,I know $\dot{\gamma}(t)=\frac{\partial \gamma}{\partial t}$, but , what is mean of $\dot{\gamma}^j(t)$? $\gamma(t)$ is a path on Riemannian manifold.
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36 views

How to understand minimising length is equivalent to energy minimising?

How to understand the equivalent above red line in picture below ?
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30 views

Smooth of metric under Ricci flow

Let $(M,g)$ is a Riemannian manifold.$g$ evolve under $\partial_tg_{ij}(x,t)=-2R_{ij}(x,t)$. When I read Shi's derivative estimate, I need the metric $g_{ij}(x,t)$ to be continuous with respect to ...
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31 views

Contraction of $(2k,2l)$tensor

In picture below ,how to get the equality in red box?
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43 views

Generally compute Gauss curvature

For a surface $F(x,y,z)=0$, we want to compute its Gauss curvature. I tried to suppose $z=f(x,y)$ locally and get a complicated expression. Is there any direct way to compute this? Thanks for your ...
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27 views

How to integrate by parts the symmetric covariant derivative on a manifold with boundary?

Let $(M,g)$ be a compact Riemannian manifold. Also let $\mathcal{S}^2=\Gamma(S^2T^*M)$ and $\mathcal{S}^3=\Gamma(S^3T^*M)$ be respectively the spaces of symmetric $2$-covariant and $3$-covariant ...
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Can we consider compact sets of Riemannian manifolds as ones of closed Rimanninan manifolds?

Let $(M,g)$ be a $C^\infty$ Riemannian manifold of $n$ dimensional and suppose $\emptyset\neq K \stackrel{\mathrm{compact}}{\subset} M$. Then are there any neiborhood ...
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44 views

A beginner's question of Riemannian Geometry.

In picture below ,I don't know why $\Phi^{-1}(F)=(F(\phi^i,e_j))_{i,j=1}^n$
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24 views

Example of fibre bundle is locally product but not globally

When I read the below picture ,I can't make a example for claim of red box.
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42 views

Can someone help me understand the Euclidean metric?

A Euclidean metric is defined as: $g_{euclid} = g_{ij} = \delta_{ij} dx^i \times dx^j = dx^1dx^1 + \ldots + dx^ndx^n$ Can someone explain the following: why do we use $dx^i$ instead of $x^i$ which ...
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65 views

Compact Einstein manifolds with $\operatorname{Ric}(g)=\lambda g$ with $\lambda<0$ and sectional curvatures $\geq0$

Does there exist a compact Einstein manifold $(M,g)$ with $\operatorname{Ric}(g)=\lambda g$ and $\lambda<0$ and nonnegative sectional curvatures?
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13 views

Higher order derivatives than Riemann Tensor

Does anyone know of any meaningful tensors that are related to the derivative of the riemann tensor? i.e. in the following picture we can consider Given arbitrary Pseudoriemannian manifold and metric ...
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33 views

Can the curvature tensor (or any symmetric tensor) be diagonalized in a non-orthogonal basis?

I have a surface embedded in 3-space. It is parametrized using a set of two Gaussian coordinates, $x^1$ and $x^2$. From the parametrization, we may determine ${\bf a}_1$ and ${\bf a}_2$, the covariant ...
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The crhonological future set of $a\in M$, $I^{+}(a)$, is open in $M$.

I'm studying some topics about Lorentz Geometry and I have a little problem in the proof of a Causality Condition theorem. Let be $(M^{n+1}, g)$ a spacetime (Lorentz manifold, connected and ...
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32 views

Path Lengths on a Sphere example.

What does $\rho$ mean as to be measured along geodesics and more importantly how would i be able to parametrize this accordingly as being on the sphere's surface? I know that I have to use the First ...
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27 views

Intersection of the unit (geodesic) sphere with the $y$-axis

I have a question about this example of Do Carmo's Riemannian Geometry. If we consider $S_1((0,1))$, the image of the unit sphere in $T_{(0,1)}G$ under the map $\exp_{(0,1)}$, at which points does it ...
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21 views

Compute of variation

How to compute the equality 2 ? I think it's to use normal coordinate,then $\Gamma_{ij}^k=0$, then $\nabla_h\Gamma_{ij}^k=0$, then $R_{ijk}^h=0$. Whether I am right ? $\delta$ is variation, ...
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Extend of cutoff function

$(M,g_t)$ is a family of Riemannian manifold ,$g_t$ evolve under Ricci flow $\partial_t g_{ij}=-2R_{ij}$. At $t=0$ ,we define $\varphi$ as below first picture . Then ,extend $\varphi=0, $ outside ...
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proof of co-area formula and level set

How to proof the co-area formula on surfaces: Let for each $t \in [0,T]$, $\phi( t,\cdot) : {\bar \Omega} \rightarrow R$ be Lipschitz continuous and assume that for each $r \in ( ...
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1answer
24 views

Geodesic Property and Parametrization of Surface.

Consider the parametrization below. How can we show that the curve $v=v_{0}= constant\,value$ are geodesics (when parametrized by arc length) ? The answer talks about the curve being contained in ...
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Geometric Structures of a fixed area.

Lets $M_A$ be the space of metrics of area $A$ on a two dimensional surface $S$, and let $D_0$ be the group of area-preserving diffeomorphisms whose right action on $M_A$ is given by pullback. The ...
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38 views

Curvature on product Riemannian manifolds

I am working on the following problem from Lee's Riemannian Manifolds: Suppose $g = g_1 \oplus g_2$ is a product metric on $M_1 \times M_2$ (i.e. $$g(X_1+X_2,Y_1+Y_2) = g_1(X_1,Y_1)+g_2(X_2,Y_2),$$ ...
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scalar curvature under conformal deformation of a two - dimensional Riemannian manifold

I am currently stuck with an identity that I'd love to derive myself. Suppose $(M,g)$ is a surface (a two - dimensional Riemannian manifold) without boundary. Let $\tilde g = e^{2u} g$ be a conformal ...
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Find parametrization for a possible “ruled” surface in $\mathbb R^4$

Let us endow $\mathbb R^4$ with a group law $\cdot$ such that the dilations $\delta_\lambda:(\mathbb R^4,\cdot)\to (\mathbb R^4,\cdot), (x_1,x_2,x_3,x_4)\mapsto (\lambda x_1,\lambda x_2,\lambda^2 ...
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43 views

Show a inequality by maximum principle .

As below picture ,how to get the inequality 3 by inequality 1 and 2 ? It seem relate to PDE, but when I use the maximum principle , I don't know how to deal the $-F^2$ . Maybe, this question is ...
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68 views

Curvature of given metric space

As my question 1 and 2, I still have many problems. First, the hyperbolic manifold is the manifold $(\mathbb R^n , g)$ given by one chart $\mathbb R^n$, where in spherical coordinates $(\theta^0= s, ...
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Volume Forms Induced by Embedding

Let $(M, g)$ be a Riemannian Manifold of dimension $d$, $g$ naturally gives rise to an invariant volume form $V_M \in \Omega^d(M)$. Let $\Sigma$ be a smooth embedded submanifold of dimension $d-1$ in ...
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What is the variations?

Sorry for my poor English ,I don't know the precise mean of variations in the below picture .I think it like coordinate translation ,but I'm not sure . Besides, if $\delta g_{ij}=v_{ij}$,how to ...
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57 views

Compute of curvature

In the answer of this question,for the given metric $$g_M = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2,$$ how to compute the curvature? Whether the hyperbolic space means $M=\{x\in R^n:x_n>0\}$? ...
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Show that $-\langle\nabla|\text{Rm}|^2,\nabla|\nabla \text{Rm}|^2\rangle\le4|\text{Rm}||\nabla \text{Rm}|^2|\nabla^2 \text{Rm}|$

Here $\text{Rm}$ is the curvature tensor. When I try to compute $$-\langle\nabla|\text{Rm}|^2,\nabla|\nabla \text{Rm}|^2\rangle\le4|\text{Rm}||\nabla \text{Rm}|^2|\nabla^2 \text{Rm}|,$$ I compute ...
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22 views

Compute of a inequality about cutoff function

How to compute the inequality with red line in the below picture ? It seems to integrate the inequality 1 , but I don't know why there is $e^{CMt}$ and where the $\int_0^t\varphi|\nabla^2 \varphi|$ ? ...
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Symmetry of Kahler metric on based loop group

The based loop group, $\Omega G$, is known to admit a Kaehler metric, given as \begin{equation} g(X,Y)=2\sum_{k>0}k\textrm{Tr}(X_{-k}Y_k), \end{equation} this is given in page 150 of Segal and ...