(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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How can i prove that the Hyperbolic space is complete by using Divergent Curves?

Let $$H_+^2=\{(x,y)\in\mathbb{R}^2:\ y>0\}$$ and consider the Lobatchevski metric on $H_+^2$: $$g_{11}=g_{22}=\frac{1}{y^2},\ g_{12}=0$$ How can one prove the completeness property of this space ...
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198 views

Levi-Civita connection

Well if $\Sigma$ is a submanifold of $R^{n+p}$ and $\{e_i,e_\alpha\}$ is orthonormal frame over $\Sigma$ where the $e_i$'s are tangent and the $e_\alpha$'s are normal to $\Sigma$. Can anyone prove ...
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3answers
310 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kähler condition implies that $\omega$ is closed with respect to $d$, so ...
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1answer
314 views

Second fundamental form proportional to the Hessian

Let $(M^n,g)$ be a Riemannian manifold and $f:M\to\mathbb{R}$ a smooth function. Then the graph $S=\{(p,f(p))\mid p\in M\}$ is a submanifold of $(M\times\mathbb{R},g+g_{\mathbb{R}})$ and carries the ...
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59 views

How can i show this Equality?

Let $M$ be a complete Riemannian manifold and $N\subset M$ a closed submanifold. If codimension of $N$ is $0$ take $q\in\partial N$ and $v\in T_qN$, where $\partial N$ is the boundary of $N$ as a ...
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121 views

Curvature of particular Riemannian metric

Let $U = \{ (x_1, \dots, x_n) \mid x_j > 0 \text{ for all } j\}$ and let $\|x\|^2 = \sum_j x_j^2$. The function $x \mapsto -\log \|x\|^2$ is strictly convex on $U$ and thus defines a Riemannian ...
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1answer
236 views

version of Bianchi identity

Let $F \to E \to M$ be a smooth fiber bundle with connection $\omega$ and curvature $R$. We can form a (graded) vector bundle by taking the complex of differential forms at each fiber. Call this ...
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2answers
341 views

Geodesic distance from point to manifold

This is question 1 of chapter 9 from Manfredo do Carmo's Riemmanian Geometry. $M$ is a complete Riemmanian manifold and $N\subset M$ a closed submanifold. $p_0\in M$ and $p_0\notin N$. Let $d(p_0,N)$ ...
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2answers
258 views

Tangent spaces at different points and the concept of connection

If $M$ is a smooth manifold and $TM$ is the tangent bundle clearly $T_pM\cong T_qM$ (as vector spaces) for every $p,q\in M$. Nobody ensures that the previous vector spaces isomorphism is natural (or ...
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0answers
57 views

About Thom theorem (representation submanifold for $H_{n-2}(M)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold. And in the Harper and ...
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329 views

About Gauss-Bonnet Theorem

The Gauss–Bonnet theorem say that: If $\Sigma \subset M =\mathbb{R}^3$ is a compact 2-dimensional Riemannian manifold without boundary, then $$ \int_{\Sigma} K = 2\pi\chi_{\Sigma}$$ where $K$ is the ...
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70 views

Is this mean curvature?

Suppose $N_t:=\partial B(p, t)\subset M^{n+1}$ be the distance sphere in a Riemannian manifold. Let $\{x_1, \cdots, x_n\}$ be a coordinate of the distance sphere $\partial B(p, t)$. Hence $\{x_1, ...
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0answers
147 views

Why does the Laplace operator extend to $L^2(X)$?

Suppose $X$ is a Riemannian manifold. Then we get a Laplace operator on $C^\infty(X)$. In most texts I see the Laplace operator extended to $L^2(X)$, but I don't see how, since it does not seem to be ...
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1k views

Is the Nash Embedding Theorem a special case of the Whitney Embedding Theorem?

The Whitney Embedding Theorem states that every smooth manifold can be embedded in Euclidean space. The Nash Embedding Theorem states that every Riemannian manifold can be embedded in Euclidean ...
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1answer
56 views

Is this Laplacian comparison?

Cheeger-Colding-Minicozzi: 1995 Linear growth harmonic functions on complete manifolds with nonnegative ricci curvature in GAFA Page 952: From Laplacian comparison, we have for $r<R$, ...
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1answer
244 views

Relationship beween Ricci curvature and sectional curvature

Let $(M,g)$ be a Riemannian manifold and assume that for all orthonormal $v,z$ the sectional curvatures is bounded from below i.e. $K(v,z) \geq C$, where $C > 0$. Is it in this case possible for ...
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2answers
41 views

Inequality about convex function

This question is arised when I study the content in the following question entitled by " lower bound of a special type of convex functions " in here. Let $f: {\bf R}^n \rightarrow {\bf R}$ be a ...
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1answer
190 views

Why does it appear that Willmore energy is always zero?

The answer is "because I'm being sloppy," but the problem is I don't know exactly where I'm being sloppy. Here's my sloppy argument: Let $M$ be a smooth compact surface without boundary in ...
8
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2answers
395 views

sign error proving product rule for the Laplacian on a product of Riemannian manifolds

Given two Riemannian manifolds $M$ and $N$, of dimension $m$ and $n$ respectively, the product manifold $M\times N$ has a Riemannian structure, and there is a Laplacian operator $\Delta_{M\times N}$ ...
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1answer
138 views

Why is a Riemannian submersion a submetry

I have encountered a conclusion in the book but I do not know how to prove it. If $F$ is a Riemannian submersion from $(M,g)$ to $(N,\tilde g)$, then it is a submetry. (A mapping $F$ is a submetry if ...
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684 views

A conformal map from a horizontal half-strip to $H$

I have seen many examples of mapping the vertical half-strip, ie $-\pi/2 \lt x < \pi/2$, $y \gt 0$ to $H$(the upper half-plane) in $\mathbb{C}$ using the transformation $f = \sin z$. Would the ...
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0answers
46 views

Complex Laplacian on 0-forms

Let $M$ be a complex manifold and $\Delta^{\bar \partial} = \bar\partial^* \bar\partial + \bar\partial\bar\partial^*$ the complex laplacian. Is it true that $\Delta^{\bar\partial} f = \Delta f$ (the ...
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1answer
136 views

Continuity of the orthogonal projection into tangent space.

Let $\mathcal M \subset \mathbb R^d$ be a smooth manifold, and for each $s \in \mathcal M$ let $T_s[\mathcal M]$ denote the tangent space of $\mathcal M$ at $s$. For each $s \in \mathcal M$ let $P_s$ ...
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1answer
277 views

isometry and exponential map

I got stuck on the following questions. Can anyone give me idea how to proceed? Suppose $M$ is a Riemannian manifold and $\phi: M \to M$ an isometry map. If $\phi(p)=p$ and $\phi(q)=q$ prove that ...
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1answer
114 views

How to conclude that a path is non-trivial element of $\pi_1(M)$

Let $M^3$ be a compact manifold. If $\mathbb{RP}^2$ is embedded in $M$. Suppose, by contradiction, that $i_\sharp: \pi_1(\mathbb{RP}^2) \longrightarrow \pi_1(M)$ is non-injective and that the normal ...
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0answers
54 views

Complex structure on the product of two complex Kaehler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kaehler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
2
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1answer
215 views

How show that a surface embedded is non-orientable?

Let be $M$ a compact $3$-manifold. If $\Sigma$ is a embedded surface in $M$, such that $\Sigma$ is homeomorphic to $\mathbb{RP}^2$. If $i: \pi_1(\Sigma) \longrightarrow \pi_1(M)$ is not injective, ...
2
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1answer
48 views

systole of space projective 3-dimensional

I'm study the papper "H. Bray, S. Brendle, M. Eichmair, and A. Neves, Area-minimizing projective planes in three-manifolds, Comm. Pure Appl. Math". (see http://arxiv.org/abs/0909.1665). Let be ...
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1answer
190 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
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25 views

Special geodesic unit speed curves on a Riemannian manifold subtending one another

Let $f(s,t)$ be a two times continuously differentiable piece of a surface, which is part of a Riemannian manifold $M$. Let $0\leq s\leq 1$ and $-\epsilon<t<\epsilon$ be given such that all ...
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1answer
88 views

Geodesics in a manifold M diffeomorphic to $\mathbb S^2$

I am now reading the book Calculus of Variations written by Jost and I encountered the following problem (in Theorem 2.3.3.): Let $M$ be a differentiable submanifold of $\mathbb R^d$ diffeomorphic to ...
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218 views

Geodesic Flow is the flow of the Hamiltonian Vector field of $|\xi|$

Let $M$ be a complete Riemannian manifold with metric $g$. The geodesic flow is the one-parameter family of diffeomorphisms $\phi_t$ on $T^*M$ defined as follows. If $\xi \in T^* M$ is a vector based ...
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2answers
267 views

An exercise in the Riemannian geometry book

If $M$ is a smooth closed $n$-dimensional Riemannian manifold which is Riemannian embedded in $\mathbb R^{n+1}$, then there exists a point $p \in M$ such that the sectional curvatures at $p$ are all ...
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1answer
102 views

The immersion of Riemannian manifold

Assume that we have a Riemannian immersion of an $n$-dimensional Riemannian manifold into $\mathbb R^{n+1}$. If $n ≥ 3$, then does $M$ necessarily have non-negative sectional curvature?(I know if ...
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1answer
108 views

The degree of Gauss map

If $M$ is an $2m$-dimensional closed orientable hypersurface in $\mathbb R^{2m+1}$, then we have a Gauss map $G:M\rightarrow S^{2m}$. I have known from my differential geometry book that ...
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1answer
223 views

Bounding the Norm of the Riemann Curvature Operator

I am having trouble with exercise 26 in chapter 2 of Peter Petersen's text "Riemannian Geometry." The exercise is stated: "Using Polarization show that the norm of the curvature operator on ...
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2answers
130 views

Reference showing that positively curved balls in surfaces have area at most $\pi r^2$

Given a Riemannian surface with nonnegative Gaussian curvature, the area of a ball of radius $r$ around any point has area at most $\pi r^2$. I have a simple proof of this in the Euclidean cone case ...
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1answer
119 views

affine patch and geodesics

" the complement of a codimension-one projective subspace of RP3(real projective space) is identifiable in a geodesic-structure manner with an affine 3-space so that the group of projective ...
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1answer
87 views

geodesics and differential equation satisfied

If $\gamma(s) = \mathbf{x}(\gamma^1(s),\gamma^2(s))$ where $\mathbf{x}$ is a coordinate patch, then what is the differential equations that $\gamma^k$ ($k = 1,2$) must satisfy if $\gamma$ is a ...
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1answer
295 views

Christoffel symbols in Differential geometry iff proof

I need help in proving that $H = 0$ for a surface iff $g_{11}L_{22} - 2g_{12}L_{12} + g_{22}L_{11} = 0.$ I think that these are the Christoffel symbols exploited in some manner and normally, I'm not ...
3
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0answers
132 views

horizontal vector in tangent bundle

I have a question about Do Carmo notion of horizontal vector (page 79). So he defines natural metric on $TM$ of manifold $M$. Now he chooses vector $V\in T_{(p,v)}(TM)$ and calls $V$ horizontal vector ...
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1answer
295 views

geodesics in differential geometry

Let gamma be a straight line in a surface M. How can we prove that gamma is a geodesic? ALl I note is that a geodesic on a surface M is a unit speed curve on M with geodesic curvature = 0 everywhere. ...
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1answer
48 views

Possible lengths of geodecics

Let $M$ be a compact manifold (w/o boundary). Suppose that there is no closed geodesics on $M$ of length precisely $C$. I am trying to prove that there is an open cover $\{U_j\}$ of $M$ and $\epsilon ...
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1answer
140 views

Some manipulations on a Riemannian manifold

Let $\phi$ be a point on a Riemannian manifold $(M,g)$ and $\xi \in T_\phi M$. Then I want to understand the proof/meaning of the following three identities I ran into, $\nabla _\nu \xi ^i = ...
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1answer
437 views

parallel transport preserves orientation

In my text its written that parallel transport on a Riemannian manifold preserves orientation. Can someone clarify what does that mean? I am confused about this notion.
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1answer
142 views

condition for compatible connection on a Riemannian manifold

Prove that connection $\nabla $ on a Riemannian manifold $M$ is compatible with metric iff $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ for every smooth vector fields $X,Y,Z$. I am confused about ...
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1answer
269 views

Total mean curvature in $L^2$ and minimal surfaces in spaces with non-positive sectional curvature

Let's suppose we have a Riemannian $n$-manifold $(N,g)$ and an immersed surface $f:\Sigma\rightarrow N$, with genus zero, equipped with the induced metric. Let's further assume that the ambient space ...
2
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1answer
44 views

What are $\partial/\partial f^j$ in Jost's definition of the differential mapping?

Let $M$ be a $d$-manifold and $x_0=(x^1,x^2,\cdots, x^d)\in M$, Jost defines the tangent space at $x_0$ to be \begin{equation}\{x_0\}\times \operatorname{span}\left\{\frac{\partial}{\partial ...
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1answer
163 views

The smoothness of distance function

$(M,g)$ is a Riemannian manifold and $N$ is a submanifold of $M$, then is the function $r(x)=\mathop {\min }\limits_{y \in N} d(x,y)$ smooth near $N$? ($d(x,y)$ is the distance function induced by ...
2
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1answer
195 views

The Riemannian metric on manifold with boundary

$(M,g)$ is a Riemannian manifold with non-empty boundary and $DM$ is the double of $M$, is there a Riemannian metric $G$ on $DM$ such that $g=i^*G$? ($i$ is the inclusion from $M$ to $DM$)