(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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222 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
297 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
229 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
55 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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296 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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68 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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135 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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947 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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53 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
209 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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40 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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184 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 ...
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362 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
123 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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633 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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42 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
122 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
256 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
105 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
51 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 ...
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1answer
202 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, ...
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1answer
46 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
178 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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24 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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206 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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254 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
87 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
98 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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210 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
124 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
118 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
81 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
290 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 ...
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115 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
288 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
47 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
133 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
410 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
134 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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258 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
154 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 ...
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1answer
164 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 $DM$)
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1answer
128 views

Necessary and sufficient condition for a vector field to be a coordinate vector field

Let $X$ be a vector field on a manifold $M$. Is there a necessary and sufficient condition on $X$ for it to be locally equal to the coordinate vector $\partial_j$ for some coordinate system? For any ...
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67 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
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1answer
258 views

Weitzenböck Identities

The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
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163 views

What are spinor fields?

For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as $$ T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}} $$ with local coordinate ...
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1answer
155 views

Volume of a Riemannian manifold and its relation to the area

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem): If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...
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1answer
56 views

Question on tensor calculation in Reimannian geometry

Given a Riemannian manfiold $M$ with metric $g=(g_{i,j})$. Let $T=T_{A,B,C,\dots}^{a,b,c,\dots}$ be a tensor on $M$. I would like to compute for example $T_{A,B,C,\dots}^{a,n,c,\dots}g_{n,m}$. We ...