(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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Isometric trivialization of tangent bundle of Riemannian manifolds

Let $(M,g)$ be a Riemannian manifold. $x\in M$. We know that there is a neighborhood $U$ of $x$ we can have a isometric trivialization $\pi$ of $TM$, Given by the orthonormal frame on $U$. Formally, ...
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90 views

derivative of a positive definite matrix

Suppose that $A$ is a positive definite symmetric matrix, specifically a Riemannian metric. Can we say anything about the sign of $tr(A^{-1}\partial_i A)$?
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1answer
45 views

Jacobi fields in polar coordinates.

This is from Sakai's Riemannian Geometry: Let $(r, \theta)$ be polar coordinates of the plane. We define a Riemannian metric $g$ on the plane by $g(\frac{\partial}{\partial r}, ...
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1answer
109 views

Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
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1answer
145 views

Euler Lagrange equation for harmonic maps

In the paper "The existence of minimal immersions of 2-spheres" by Sacks and Uhlenbeck the authors claim that the Euler Lagrange equation for the modified functional $E_\alpha(s) = \int_M (1 + ...
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1answer
57 views

$2$-dimensional Hyperbolic space with fundamental group ${\bf Z}$ and constant curvature $-1$

$$ d\rho^2 + \cosh^2\rho\ d\theta^2$$ Only one ? Is there any other example ?
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23 views

Orthogonal irreducible decomposition of $\otimes^2 E$

Recall $$ \otimes^2 E = \wedge^2 E \oplus S^2_0E\oplus {\bf R}$$ Clearly this is $O(n)$-decomposition. Irreducibility can be checked from the following property : Let $Ae_1=e_k,\ Ae_2=e_l,\ ...
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20 views

Decomposition of $S^2(\wedge^2 E)$

Consider bianchi map $$ b(T)(x,y,z,t) = \frac{1}{3}(T(x,y,z,t)+T(y,z,x,t) + T(z,x,y,t))$$ where $T\in S^2(\wedge^2 E)$ I already checked that $b(b(T))=b(T)\in S^2(\wedge^2 E)$ But how can we derive ...
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1answer
56 views

Fundamental group of a component of $GL_n({\bf R})$

Let $G$ be a component of $GL_n({\bf R})$ such that element has a positive determenant. (1) Since it contains $SO(n)$, $\pi_1(SO(n))$ ? What is a fundamental group of $G$ ? (2) It has a curvature ...
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69 views

Observation on normalized Ricci flow on two sphere

Note that two sphere with nonnegative curvature converges to canonical sphere along normalized Ricci flow. So assume that if curvature is bounded below by $-\epsilon$ then the sphere meets a ...
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1answer
33 views

Exponential map on $SO(3)$

(1) As I read some article in here ( I cannot found ), so we know that $$ {\rm exp} \ (T_eSO(3)) \neq SO(3) $$ ( ${\rm diag}(-1,-1,1)$ cannot be covered by ${\rm exp}$ ) But there exists some open ...
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107 views

A linear connection induces a covariant derivative of tensor fields.

Let $M$ be a smooth manifold. notation: $\mathcal T(M)^{(k,l)}$ is the $C^{\infty}(M)$-module of all tensor fields of type $(k,l)$ on $M$ ($k$ indicates the covariant part). $\mathcal ...
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1answer
73 views

Parameterize a geodesic using one of the coordinates

I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the ...
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1answer
51 views

Canonical projection of tangent space onto the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold, without fixing metric, nor parallel transport). Let $x\in S^1$. Then it seems to me that there is no canonical projection $\pi : T_x S^1 ...
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1answer
34 views

Visualization of locally product Riemannian metric.

If $S^1\times M$ where $M$ is a simply connected compact manifold has a metric $g$ with nonnegative sectional curvature, then its universal cover ${\bf R}\times M$ has a product metric by splitting ...
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40 views

O'Neill Formula in terms of Exterior Derivative of Killing Form

O'Neill Formula : Consider a fibration $\pi : (M ,g)\rightarrow M/G$ where $G$ has only one orbit type. Then we have $$ K_{M/G} (d\pi V, d\pi W) = K_M(V,W) + \frac{3}{4} | [V,W]^V |^2_g$$ where $V,\ ...
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1answer
101 views

What groups can be realized as the isometry group of the two-sphere?

Regarding $S^2 \subseteq \mathbb{E}^3$ as a Riemannian manifold with the inherited metric from Euclidean three-space, then it is well known that the isometry group is $O(3)$. What I am curious about, ...
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60 views

French translation, and what is the curvature of a metric?

I have a french paper to read. There is the notion of une collection des courbures des métriques $g_t$. Now I would guess that this refers to a collection of curvatures of metrics $g_t$, however ...
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1answer
38 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
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161 views

Compatibility of a connection and metric

Every Riemannian manifold admits a metric connection. Suppose $M $ is a manifold and $\nabla $ is an arbitrary connection on the tangent bundle. Does $M$ necessarily admit a metric such that $\nabla$ ...
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95 views

Metric Tensor on Lie Group for Left Invariant Metric

Let $G$ be a Lie group and $Q$ be a biinvariant metric. If $h$ is any positive definite scalar product on $T_eG$ then we have a left invariant metric $h$ on $G$ : $$ h_g (dL_g X, dL_g Y) = ...
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2answers
64 views

Question about conformal maps.

By definition, a diffeomorphism $\sigma:(M,g)\to (N,h)$ is called conformal if $\sigma^*h=ug$. Another definition I've seen in other contexts is that conformal maps are ones that preserve angles. Now ...
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64 views

Spivak vol. 2 — expression of Riemann's quadratic function

I would very much appreciate it if someone could explain or at least indicate a proof of the following assertion in Spivak's ''Compr. Intro. to Diff. Geom.'', vol. 2, p. 171 (3rd ed., 2nd printing): ...
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136 views

The distance function of the geodesically convex manifold

$M$ is a geodesically convex Riemannian manifold, that is, for any two points $p,q$ on $M$, there is a unique minimizing geodesic connecting them. Can we conclude that for any $p \in M$, the function ...
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83 views

Ambrose Singer Theorem

I wish to learn about holonomy groups of Riemannian manifolds and the Ambrose- Singer theorem. Please advise some references other than the original paper of Ambrose and Singer.
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180 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
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57 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
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3answers
78 views

Corollary to Preissman's theorem

Preismann's theorem states (ref. Petersen's "Riemannian Geometry", chapter 6): On a compact manifold with negative sectional curvature, any abelian subgroup of the fundamental group is cyclic. A ...
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1answer
121 views

locally isometric is not a symmetric relation.

The relation of being locally isometric for Riemannian manifolds is reflexive and transitive. Is it symmetric? Can you give me an example?
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36 views

mean curvature and volumes of submanifolds

How to relate the mean curvature vector evolution over a submanifold of an euclidean space to growth of the volumes of geodesic balls. Can i determine the volume of a geodesic ball by integrating the ...
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1answer
59 views

Radial geodesics in a graph of a function

I'm trying to figure out how to prove the following claim: Suppose that $S$ is the graph of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and every plane containing the $z$-axis intersects $S$ ...
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43 views

Sobolev diffeomorphisms.

Let $M$ be a compact Riemannian manifold without boundary. Suppose $f \in H^s(M,M)$, where $H^s$ denotes the ($L^2$-based) Sobolev space. Assume $s > n/2 + 1$, so that by the Sobolev Embedding ...
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61 views

Geodesic on a Reimannian manifold with a random metric tensor

Given a metric tensor $g_{\mu\nu}$ on a Riemannian manifold, it's possible to write the geodesic equations using: $$\frac{d^2x^a}{ds^2} + \Gamma^{a}_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds} = 0$$ where: ...
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49 views

Addition of Fundamental Vector Fields

If we define a fundamental vector field, i.e., $$ X^\ast =\frac{d}{dt}|_0 \exp(tX)\cdot p $$ where $p\in M=G/K$, Question 1 : then for $X,\ Y\in (T_eK)^\perp$, we have $$ X^\ast + Y^\ast = ...
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302 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
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0answers
25 views

subharmonic function and support functions

$M$ is a Riemannian manifold and $f$ is a continuous function on $M$. $f$ has the property that for any $p \in M,\epsilon>0$, we can find a smooth function $f_{\epsilon}$ such that ${f_\varepsilon ...
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37 views

Explanation required of the following definition:

This is a definition I encountered in a paper. I hope someone will be able to help me understand it. The authors assume a Frenet curve $\alpha(s)$ on a 3-D Riemannian Manifold as any non-geodesic unit ...
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2answers
187 views

An Einstein manifold has constant scalar curvature.

I know this is called Schur's lemma. But I cannot find a proof. All references available to me either does not give a proof, or says that it is similar to the lemma for sectional curvature, making use ...
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1answer
65 views

Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
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57 views

Question from Peter Petersen

I'm trying do a exercise from Peter Petersen's book, but I don't know what do. Well, assume that $$R=\frac{scal}{2n(n-1)}g\circ g+\left(Ric-\frac{scal}{n}g\right)\circ g+W$$ Where, R is the ...
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98 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
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44 views

Gauss-Bonnet Theorem in dimension four

I've read that the generalized Gauss-Bonnet theorem states that $$\int\limits_{M}Pf(\Omega)=(2\pi)^n\chi(M)$$ where, $M$ is a 2n-dimensional compact orientable Riemannian manifold without ...
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1answer
106 views

Need help finding a good book on Riemann Geometry

I want to learn more about calculus on manifolds and Riemann Geometry. I have been reading the book Geometry, Topology and Physics by Nakahara. But I find that it is difficult to read due to the lack ...
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236 views

Lie bracket is a connection?

In Road to Reality, section 14.6 on Lie derivative Penrose writes: Now $\epsilon^2 [j,h]$ corresponds to an $O(\epsilon^2)$ gap in the ‘parallelogram’ whose initial sides are $e_j$ and $e_h$ at ...
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1answer
162 views

Covert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels x units in Euclidean space, how much ...
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1answer
169 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
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1answer
103 views

Metric on Riemannian manifolds

Why is it necessary to consider taking the infimum over the lengths of all piece-wise smooth curves while defining the distance function on a Riemannian Manifold instead of just taking the infimum ...
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37 views

Diagonalization of Curvature Operator of $P^n({\bf C})$

Consider $P^n({\bf C})$ which is a quotient of $(S^{2n+1}, {\rm can})$. If $ \{e_1, ... , e_n, Je_1, ... , Je_n\}$ is a basis on $T_xP^n({\bf C})$ where $J$ is an almost complex structure, then ...
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58 views

bounds on eigenvalues of elliptic operators on functions on riemannian manifolds

Well I have little experience with pde's and analysis, I mostly study topic related to geometric topology and I would like to see if someone can please explain to me why is it important to find bounds ...
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174 views

Creating geodesics on manifolds

Suppose I have two points on a Riemannian manifold $M$, called $p_0$ and $p_1$. I have a family of curves $\gamma:[0,\infty)\times[0,L]\to M$ such that $\gamma(t,0) = p_0$ and $\gamma(t,L) = p_1$. ...