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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frame homogeneous manifold

A semi riemanian manifold M is frame homogeneous provided any frame on M can be carried to any other by differential map of an isometry of M. How do I prove a flat connected frame homogeneous manifd ...
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Mathematica packages for symbolic computations of Christoffel symbols and parallel transports in Riemannian gaometry,

I've no knowledge in mathematica (but I do in matlab), but I'd really appreciate if someone could mention what is/are the best and easy to learn mathematica package(s) for symbolic and numerical ...
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30 views

a question about differential geometry(Gauss-bonnet theorem and isolated singular point in the surface)

Let C be a regular closed simple curve on a sphere $S^2$. Let v be a differentiable vector field on $S^2$ such that the trajectories of v are never tangent to C. prove that each of the two regions ...
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33 views

a question about undergraduate-level differential geometry(Gauss-Bonnet theorem)

Let $S\subset R^3$ be a regular surface homeomorphic to a sphere. Let $\alpha\subset S $ be a simple closed geodesic in S,let A and B be a regions of S which have $\alpha$ as a common boundary. Let ...
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15 views

When are heat kernels only dependent on the distance?

"All" the examples of heat kernels in circulation are only dependent on the distance between the space variables rather than on the space variables themselves, i.e. $$K(t;x,y) = K(t;d(x,y)).$$ Think ...
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24 views

Geodesic parameterization under conformal mapping

Under a conformal deformation of the euclidean metric, say: $\hat{g}_{ij}=e^{\phi}\delta_{ij}$, where $\phi$ depends on the radial coordinate alone, I am struggling to see the following fact: "With ...
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32 views

Verification of the identity $\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$

In the book Riemannian Geometry, page 91, Do Carmo writes: $$\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle$$ I could not understand how this happens. Can someone ...
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57 views

Is it possible to compute geodesic without induced metric

Suppose a manifold embedding $i:M\to N$ into Riemannian manifold $(N,g)$ is given by $f(x)=0$, where $f:M\to R^m$ is a smooth vector-valued function. Now if it is very hard to parameterize the ...
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23 views

Does every smooth manifold admits non-isometric riemannian metrics?

I want to get a feeling for how much flexibility we have when putting a Riemannian metric on a given smooth manifold $M$. Is it always possible to find two non-isometric metrics on $M$? If the ...
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13 views

Lift of Ricci curvature: problem in a proof

I am reading a proof in the book $\textit{Semi-Riemannian Geometry}$ from O'Neill, and there is a step which confuses me. The lemma is the following: $ M=P \times_r S^2$ ($P = \mathbb{R}\times ...
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45 views

What intuition do we have for a subalgebra of Lie to be abelian?

The motivation for my question comes from the definition of rank of a given globally symmetric space: it is based on the image of a maximal abelian subalgebra of a given algebra by the exponential ...
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Self contained book on Ricci Flow/Geometric Analysis

Can someone please tell me whether there is any self contained book on Geometric Analysis/Ricci Flow/analytic techniques used in Riemannian Geometry? By self contained I mean it does not assume that ...
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19 views

Necessary of completeness assumption for Cartan Hadamard theorem

I have learnt the Cartan Hadamard theorem, Let $M$ be a complete Riemannian manifold with nonpositive sectional curvature. Then $\forall x\in M, \exp_x:T_xM\to M$ has no conjugate point. Then the ...
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1answer
38 views

Induced Connection on $\Sigma\subset M$

Let $(M,g)$ be a Riemannian manifold, $\Sigma$ a manifold and $F:\Sigma \rightarrow M$ a smooth map. For $X,Y \in \Gamma(T\Sigma)$ vector fields and $\tilde{\nabla}$ the pull back connection on ...
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is every totally geodesic submanifold the set of fixed points of some isometries?

It is well known that the set of fixed points of an isometry $\phi:(M,g)\rightarrow (M,g)$ is a totally geodesic embedded submanifold. (e.g here ). I ask whether the converse is true, i.e is every ...
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40 views

Question about parallel displacement on a surface

This is Problem 9.6(1) from the book The Geometry of Physics: What's wrong with the following argument? A vector $\mathbf v$ is parallel displaced around a small closed curve $C = ...
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20 views

Unbounded Geodesics and Nonpositive Curvature

I have the following interesting(?) question: Let $M$ be a complete Riemann manifold. Suppose that it is of non-negative curvature. I want to know if $M$ has unbounded geodesics. As the question is ...
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product metric on Riemannian manifolds

Let $M_1$ and $M_2$ be Riemannian manifolds and consider the cartesian product $M_1 \times M_2$ with the profuct structure .Let $\pi_1: M_1 \times M_2 \to M_1$ and $\pi_2: M_1 \times M_2 \to M_2$be ...
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23 views

Existence of a particular vector in kernal of $d \exp_x$

I want to prove $(d\exp_x)_{p}$ is singular iff there exists a normal Jacobi field $U(t)$ along $\gamma(t)=\exp_x(tp)$ not identically zero such that $U(0)=U(1)=0$. I have question about ...
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1answer
34 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
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59 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
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42 views

Different definitions of tangent vector

I'm taking general relativity at the moment, and today in class the instructor gave us a definition of tangent vector as: $v$ is a tangent vector based at $p\in M$ if $v_{p}$ is a linear ...
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1answer
39 views

To calculate covariant derivative in Riemanian Geometry

I want to solve the exercise $2.57$ using $2.56$ I know calculate $2.57$ by using christoffel symbols but this process is long.How can I solve this directly via $2.56$. Can somebody help me by ...
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32 views

How do you get the curvature tensor of the Schwarzschild Solution?

So, on the Wikipedia page on the derivation of the Schwärzschild solution , I get everything up to the part about the Ricci tensor. What were the components of the tensor that were used? Could ...
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17 views

CAT(K) Finsler manifolds.

I was wondering if the following is true (and common knowledge): Let $(M,F)$ be a Finsler manifold. Let d be the induced distance by the norm in the usual sense. That is, $d(x,y)=\inf${lenghts of ...
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1answer
19 views

A triangle inequality for angles

Let $M$ be a complete Riemannian manifold with nonnegative curvature and $x,y,z,p$ four points on $M$. We denote by $\theta(x,y),\theta(y,z),\theta(z,x)$, respectively, the angles at $\tilde p$ of the ...
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42 views

Gradient in local coordinates on a manifold with Riemannian metric

Let $M$ be a smooth manifold with a Riemannian metric g : $TM\otimes TM$ -> R If f is a smooth function from M to R, the gradient of f with respect to g is the vector field $\nabla f$ defined by ...
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1answer
49 views

Why is $\frac{d}{dt}|\xi(t)|^2=2<\xi, \nabla \xi>$, when $\nabla$ is the covariant derivative?

I understand that: $(1) \frac{d}{dt}|\xi(t)|^2=2<\xi, \nabla \xi>$ if this is happening in $\mathbb{R}^n$. But my question is in the following setting: $\nabla \xi:=K_{c^*\pi} \circ T ...
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2answers
44 views

Exterior differentiation of one form on a smooth manifold

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I'm fine with the right side of the equation, ...
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1answer
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Converse of statement related to Hopf-Rinow Theorem

I know the Hopf-Rinow theorem and that if a Riemannian manifold is complete it implies that given any two points there is a unique distance minimizing geodesic that connects the two points, but is the ...
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44 views

Extension of isometries on submanifolds of a riemannian manifold

Let $S$ a submanifold of a riemannian manifold $M$ such that the closure of $S$ is equal $M$. i.e $\bar{S}=M$, when can we extend an isometry(as riemannian manifold) $f:S\to S$ to an isometry ...
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1answer
39 views

Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and ...
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63 views

Coordinate expression for the divergent

Let $(M, g)$ be a Riemannian manifold. As in Lee's Riemannian Manifolds book, we define the divergent of a vector field $X \in \mathfrak{X}(M)$ by the identity $d(\iota_X dV) = (div X) dV$, where ...
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40 views

Expression for codifferential in terms of interior product

Let $(M^n,g)$ be a Riemannian manifold with local orthonormal frame $\{e_1,\ldots,e_n\}$ with dual basis $\{e^1,\ldots,e^n\}$ and with Levi-Civita connection $\nabla$. It can be checked on basis that ...
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26 views

a question about finding umbilical points in an elipsoid.

Determine the umbilical points of the elipsoid $${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$$ My thoughts: let $x=asin(\theta)cos(\phi),y=bsin(\theta)cos(\phi),$and $z=cos(\theta)$. Thus, I ...
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a question about differential geometry

Let $S$ be a surface and $x: U\to S$ be a parametrization of $S$. If $ac-b^2 <0$, show that $$a(u,v)(\dot u)^2+2b(u,v)\dot u\dot v+c(u,v)(\dot v)^2=0$$ can be factored into two distinct equations, ...
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32 views

Sets that are convex in two different metrics

Let $(M,g)$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining ...
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1answer
33 views

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result: I have three questions: (1) Is $h$ also a conformally flat ...
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16 views

metric reversed on $S(n,v)$

On page 204 of O'Neill, Barrett (1983), Semi-Riemanniann geometry warped product is explained. How can I calculate the metric tensor on $S(n,v)$ where $v$ is the index of $S$ and $n$ is its ...
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19 views

All rank two symmetric tensors are several conformally flat metrics summed together?

If given a rank two symmetric tensor $T_{mn}$ can I decompose it as \begin{align} T_{mn} = \sum_{i=1}^{M} \phi_ig_{mn}{}^{i}{} \end{align} where $\phi_i$ are the $i$th conformal factors and ...
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1answer
46 views

Notation in symmetric and alternating products of forms

In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric ...
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$\Delta_L(\text{im}\,\delta^*_g)\subset\text{im}\,\delta^*_g$ and $\Delta_L\big(\text{ker}\,\text{Bian}(g)\big)\subset\text{ker}\,\text{Bian}(g)$?

Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The Lichnerowicz Laplacian of $g$ is the map ...
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How to prove the skew symmetry of the Riemannian tensor?

I parallel transported a vector around a parallelogram formed by another two vectors to get the Riemannian tensor: $$R^m{}_{lkj} = \left( \partial_k \Gamma^m{}_{jl} + \Gamma^m{}_{kn} \Gamma^n{}_{jl} ...
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1answer
36 views

Kähler–Einstein condition

Let $(M,\omega)$ be a Kähler manifold, and $g$ the Riemannian metric such that $\omega(X,JY)=g(X,Y)$. If there is a function $f$ such that $\operatorname{Ric}=f\omega$ does this mean that ...
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28 views

Dilating a curved ball

Let $B$ be a ball sitting inside a manifold $(M^n, g)$. Now, let us dilate the metric $g$ to $\lambda g$, $\lambda$ being a positive number going to $\infty$. It seems intuitively true that the ...
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26 views

Adjoint of Exponential Map

If $\exp: T_p(G) \rightarrow G$ is the expoenential map of a lie group, then what does the adjoint operator (as in $\langle Ax,y\rangle=\langle x,A^*,y\rangle$) of the derivative of exp look like? ...
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77 views

What is the geometric interpretation of the Koszul formula?

I saw this simple form of Koszul formula on a book: $$2\ g(\nabla_XY,Z) = \mathcal{L}_Yg(X,Z) + (d\theta_Y)(X,Z)$$ where $\theta_Y$ is the one-form $g(Y,\cdot)$. It is equivalent to the more ...
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1answer
43 views

Lower bound on convexity radius in terms of injectivity radius (without using curvature)

Let $M$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining $p$ ...
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41 views

Differential Geometry - Distributions mutually orthogonal, span the tangent space, parallel imply manifold splits locally as product manifold

I'm stuck on a portion of Exercise 21, Chapter 2 in Petersen's Riemannian geometry text. Fix a Riemannian manifold $(M,g).$ Suppose that I have two distributions $D^1$ and $D^2$ defined on $M.$ ...
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50 views

Geodesics on Lorentzian (2n-1)-Spheres

I know that if we endow $S^{n}$ with the round Riemannian metric, we will be able to join the North pole and the South pole by an unlimited number of geodesics, in particular the meridians, and indeed ...