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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107 views

How much classical geometry must a geometer know?

From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know): Absolute Euclidean Non-Euclidean ...
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61 views
+50

An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that $\gamma'_1(1)...
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1answer
10 views

Are segment domains closed?

Let $M$ be a complete Riemannian manifold. Its segment domain is defined by: $$ \mathbf{seg}(p)= \{v\in T_pM: \exp_p(tv):[0,1] \to M \textit{ is a segment} \ \ \} $$ (Note: "segment" has many ...
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26 views

Full definition of Rough Laplacian and induced formal adjoint of covariant derivation?

Can everybody give a good reference for full definition of Rough Laplacian of tensor field and induced formal adjoint of covariant derivation on a riemannian manifold? I find some equivalent ...
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40 views

Why Bi invariant metric on noncompact lie group doesn't exist??

In the book "Lectures on Differential Geometry" by Sternberg page 233 "Given a representation,p, of a Lie group G (in particular the adjoint representation) on a vector space F, if p(x) is compact ...
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2answers
35 views

Why image of curvature is a Lie subalgebra?

In the red line of picture below, why it is Lie algebra ? $M_{\alpha\beta}$ is the Lie bracket ? But $M_{\alpha\beta}$ is symmetric . Picture below is from the 216 page of this paper.
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27 views

Proof of Hamilton's strong maximum principle.

As picture below, Why $\forall v\in \text{null}(M_{\alpha\beta}), \nabla_iv\in\text{null}(M_{\alpha\beta})\Rightarrow \text{null}(M_{\alpha\beta}) \text{ is invariant under parallel translation}$ ? ...
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33 views

Rank of curvature operator under Ricci flow.

I think under Ricci flow ,the rank of curvature operator does not change by +1 or -1, it will directly change to full rank or zero rank . I want to write it as term paper, but I don't know whether ...
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1answer
51 views

Local picture of a Riemannian manifold with constant sectional curvature.

Theorem: If a Riemannian n-manifold $(M, g)$ has constant sectional curvature $k=1$, then every point in $M$ has a neighborhood that is isometric to an open subset of the space form $S^n$. (cf. ...
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0answers
32 views

does a vector field and its dual 1-form over an n-sphere form the heisenberg group?

Apologies, I don't claim my reasoning is perfect, but I would appreciate any critiques. Thank you. Let us consider commutation relations on a general Riemannian manifold M , where the commutator ...
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0answers
33 views

Length of a differentiable curve with respect to a Riemannian metric.

Let $X$ be an $n$-dimensional differentiable manifold ($n\ge1$). A Riemannian metric in $X$ is a family $\{g_p\,|\,p\in X\}$, where for all $p\in X$: $g_p:T_pX\times T_pX\to\mathbb{R}$ is an inner ...
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1answer
24 views

Calculating the second fundamental form of surfaces

I am asked to prove that for a surface in $\mathbb{R}^3$ with local coordinates in a chart, u,v, the coefficients of the second fundamental form can be calculated as follows: eg.: (first entry in II) ...
3
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1answer
42 views

Question about two ways to induce an inner product on $S^2V$

$\newcommand{\til}{\tilde}$ Let $(V,g)$ be an $n$-dimensional inner product space, and let $S^2V^*$ be the symmetric algebra. I am familiar with a natural way to endow $S^2V^*$ with an inner product ...
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3answers
28 views

Simple question about geodesic in Riemann manifold

I was read my book of Riemannian Geometry and the book says the follow: " A parameterized curve $\gamma:I\to M$ is a geodesic in $t_{0}\in I$ if $\dfrac{D}{dt}\left(\dfrac{d\gamma}{dt}\right)=0$ in ...
12
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1answer
324 views

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
1
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1answer
33 views

Orientable on almost complex manifold

I have troubles trying to prove almost complex two-dimensional manifold is orientable. Let I is complex structure on two-dimensional manifold M. Fix a basic $X_1,IX_1$ in each $T_xM$. Easy to see ...
4
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1answer
78 views

The cylinder does not embed into $\Bbb C^n$

The cylinder $\Bbb R\times S^1$ can be viewed as a complex manifold with a flat metric by viewing it has the quotient $\Bbb R\times\Bbb R/\Bbb Z$, where $\Bbb R\times\Bbb R=\Bbb C$. (In fact it makes ...
0
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0answers
40 views

Inclusions maps, parameterisation and charts

So it makes sense that an inclusion map $$\iota : S \longrightarrow S \subset M $$ maps $$ p \mapsto p $$ But how do you construct these guys in the context of manifolds? To my understanding, if $S$ ...
3
votes
1answer
62 views

Parallel transport along a closed geodesic

It do Carmo, in exercise 9.4, it is claimed that parallel transport along a closed geodesic in an even-dimensional orientable manifold "leaves a vector orthogonal to the geodesic invariant." So, let $...
3
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1answer
54 views

No conjugate points on $S^1\times \Bbb R$

Lee claims in his book that $S^1\times \Bbb R$ (considered as a submanifold of $\Bbb R^3$) admits no conjugate points along any geodesic. I am struggling to make that rigorous. Being conjugate along ...
3
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0answers
24 views

Isometries of the canonical left invariant metric on $GL_n$

Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: \...
2
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1answer
26 views

Informations about the cut-locus of a closed geodesic

Let be $(M^2,g)$ a closed riemannian manifold and $c:[0,L]\to M$ a simple closed geodesic on $M$. For each $s\in [0,L]$, let be $n(s)$ a unit normal vector field along to $c(s)$ and $\beta(s)$ the cut ...
2
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1answer
52 views

Eigenvalues of shape operator and of curvature on second exterior power

Terminology note In the following, a scalar product will be a symmetric bilinear form, and a euclidean scalar product will be a positive definite scalar product. This is the terminology used by my ...
6
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2answers
58 views

On the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$

Let $\exp$ be the exponential map on the Riemannian manifold M and $O$ is its domain in $TM$. Consider the map $E: O\subset TM \to M \times M$ by $E(v)=(\pi(v), \exp(v) )$, where $\pi$ is the ...
12
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7answers
377 views

On a definition of manifold

In the book Mathematical Masterpiece, on page 160, the authors wrote that A manifold, in Riemann's words, is a continuous transition of an instance I know a manifold is something glued by ...
7
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2answers
65 views

Three dimensional spherical excess formula

We all know the spherical excess formula: in a unit sphere, the area of a geodesic triangle is equal to the exceeding from $\pi$ of the sum of the three angles of the triangle. Is there a similar ...
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1answer
36 views

Isometric action on $S^n$

Let $S^n$ be the n dimensional sphere. For $n=2k+1$ odd, we identify $S^n$ as subset of $\mathbb{C}^{k+1}$. Furthermore we can define the action $$\Psi: S^1 \times S^n \to S^n, (c,(z_0, \dots, z_n)) ...
5
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1answer
48 views

Scalar curvature of metric? [closed]

The metric components in a two-dimensional spacetime are given in terms of the coordinates $(t, x)$ by$$ds^2 = f(x)\,dt^2 + dx^2.$$What is the scalar curvature, $R$, of this metric?
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1answer
29 views

If $M_{\alpha\beta}\ge0$ , how to show $M^\#\ge0$?

We say $M_{\alpha\beta}\ge0$ if $M_{\alpha\beta}v^\alpha v^\beta\ge0$ (sum over $\alpha,\beta$) for all vectors $v=\{v^\alpha\}$. If $M_{\alpha\beta}\ge0$ ,how to show $$ M^\#\ge0 ~? $$ Relative ...
2
votes
1answer
29 views

Identity surrounding Killing vector field on a spacetime $\nabla_a \nabla_b w_c = -{R_{bca}}^d w_d$

Let $w^a$ be a Killing vector field on a spacetime $(M, g_{ab})$, i.e., $w^a$ satisfies $\nabla_{(a}w_{b)} = 0$. I hypothesize that$$\nabla_a \nabla_b w_c = -{R_{bca}}^d w_d,$$but I am not sure how I ...
4
votes
1answer
54 views

“Flow lines” of “dust” are geodesics?

The stress-energy tensor representing "dust" takes the form$$T_{ab} = \rho u_au_b$$where $u^a$ is a unit timelike vector field, i.e., $u^au_a = -1$. Does it necessarily follow that in any solution to ...
1
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1answer
23 views

Find a isometric immersion of the Torus $T^{n}$ on $\mathbb{R}^{2n}$.

Find a isometric immersion of the "plane" Torus $T^{n}$ on $\mathbb{R}^{2n}$. Isometric Immersion, let $f:M^{n}\to N^{n+k}$ a immersion, i.e., $f$ is differentiable and $df_{p}:T_{p}M\to T_{f(p)}N$ ...
2
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1answer
57 views

About totally umbilical hypersurfaces

Suppose $\tilde{M} \subset M$ is a hypersurface sitting inside a Riemannian manifold $(M,g)$. The second fundamental form of $M$ evaluated on $u,v \in T_pM$ is denoted $II(u,v)$ and defined as the ...
2
votes
1answer
36 views

Covariant Contravariant approach for Tensors

I'm reading a book on Geometry from the '70s and when speaking about Tensors it defines them starting from the covariant and contravariant commutation rule. I know this definition was quite widespread ...
3
votes
1answer
117 views

Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
2
votes
1answer
38 views

Minimizing curves are geodesics

Let $(M,g)$ be a Riemannian manifold. I want to prove the following claim: Let $c:[0,1]\to M$ be a smooth curve from $p$ to $q$ such that $L(c)=d(p,q)$. Then $c$ is, up to reparametrization, a ...
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0answers
26 views

Isometries between the hyperboloid and the plane?

What is the map from $\mathbb{H}^2$ to $\mathbb{R}^2$ that preserves the pairwise geodesic distances in one as closely as possible to the pairwise geodesic distances of the images in the other? (......
10
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1answer
114 views

Identity in general relativity, not sure if true or not

Let $(M, g_{ab})$ be a spacetime and define a new metric, $\tilde{g}_{ab}$, on $M$ by $\tilde{g}_{ab} = \Omega^2 g_{ab}$, where $\Omega$ is a smooth, positive function. Let $\nabla_a$ denote the ...
0
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1answer
40 views

Using Koszul's formula to compute $\nabla_X Y$, where $\nabla$ is the Levi-Civita connection

Main question: I am following the answer in this question: computing Riemannian connection and Killing fields (very basic). He calculates $$ g(\nabla_X(Y),X) = 0 \\ g(\nabla_X(Y),Y) = 0 \\ g(\nabla_X(...
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1answer
45 views

Does harmonic decomposition preserve immersions?

In a nutshell: If we start from an immersion, and look at its harmonic decomposition, are the components also immersions? Details: Let $(M,g)$ be an $n$-dimensional, compact Riemannian manifold with ...
6
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1answer
26 views

Does it necessarily follow that the integral curves of $k^a$ are null geodesics?

Let $f$ be a function on a spacetime $(M, g_{ab})$ whose gradient, $k_a = \nabla_a f$, ie everywhere null, i.e., $k_ak^a = 0$ throughout $M$. Does it necessarily follow that the integral curves of $k^...
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55 views

Showing that a specifik structure on $\mathbb{R}^2$ is a complete metric

I have smooth maps $f,h:\mathbb{R}\rightarrow (0,\infty )$ with $f(t)\geq k$ and $h(t)\geq \frac{1}{\mid t\mid}$ for all $\mid t\mid >c$ for some$k,c>0$. I want to prove that $$g=f(t)ds^2+h(t)...
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1answer
39 views

When are the minimizing geodesics of a totally geodesic submanifold also minimizing in the underlying manifold?

Asked here too: http://mathoverflow.net/questions/235178/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz A reference on totally geodesic submanifold (TGS): http://...
2
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0answers
50 views

Injectivity Radius of Surface Level

Can tip this problem. I did a part , but could not complete. They gave me a tip to complete , but could not. I thank the help . $\mathbf{Problem}$ Let $\, \, \, f: \mathbb{S}^{n+1} \rightarrow \...
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0answers
43 views

Riemannian metric on a level set of a smooth function on a manifold

Also asked here: http://mathoverflow.net/questions/235163/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold Let $(M,g)$ be a finite or infinite dimensional Riemannian manifold. Let $...
2
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1answer
50 views

Computing Sectional Curvature on Hyperbolic Plane

For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as $$K_p(X,Y)=\frac{<R(X,Y)Y,X>}{|X|^2 |Y|^2 - <X,Y>^2}$$ The problem I'm ...
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1answer
23 views

Condition for a one-parameter family of maps to be isometries

Given two smooth manifolds with Riemannian metric $(X,g)$ and $(Y,h)$ and a smooth map $f: X \to Y$ I understand that we define $\phi$ to be an isometry if $f^* g = h$. I thought I understood this ...
7
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2answers
80 views

Algebraic topology & Riemannian geometry project idea?

I'm taking a first course on Riemannian geometry this semester. For a final project, I would like to do something that involves algebraic topology. However, the only results I know in algebraic ...
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0answers
43 views

Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
3
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1answer
44 views

Laplacian of a distance function on a Riemann manifold

For some reasons I need to show the following fact. Let $(M, g)$ be a Riemannian manifold. Let $U \subset M$ be an open set and $r: M \to \mathbb{R}$ a smooth distance function. Let us assume ...