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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Repost (question refined): Maurer-Cartan trivializations and Bi-invariant Metric on Lie Group

Let $H$ be a Lie group. Let $\nabla^L$ and $\nabla^R$ be the connections on $TH$ corresponding to the trivial connection $d$ on $H\times\mathfrak{h}$ under the left and right trivializations: ...
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1answer
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Local expression of hermitian metric

I have really hard times reading Zheng's Complex Differential Geometry and I find the following sentence especially baffling (sec. 7.4, page 170): "Let $M^n$ be a complex manifold. A Hermitian metric ...
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Warped products and Ricci curvature

I am studying warped products. If we consider the warped product $M = P \times_r S$, is the following statement true and if yes, why ? "Because leaves are totally geodesic, $^P\text{Ric} = ...
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1answer
38 views

Index notation.

I have a very basic question concerning index notation, normally used in physic papers. I have never used index notation and it is very difficult to me to translate from index free notation, even in ...
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36 views

Mean Curvature Flow - First Variation Formula

I'm just going through Huisken's paper, flow by mean curvature of convex surfaces into spheres, and I'm stuck on one of the Lemma's proven in the paper. The idea is to prove (for the normalised ...
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How does the Schrodinger's potential transformer if the metric conformally transformers?

Given Schrodinger's equation $$ (-\nabla^2+V)\psi=E\psi $$ and the conformal transformation $\tilde{g}_{mn}=e^{2\phi}g_{mn}$, how does the Schrodinger's potential $V$ transformer if the metric ...
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Reversal metric [on hold]

The operation of changing the semi-Riemannian manifold $M$ with metric tensor $g$ to the same smooth manifold with metric tensor $-g$ is called reversing the metric of $M$. I don't understand it. ...
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33 views

Diameter of the Grassmannian

Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...
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1answer
39 views

Riemannian metric of $3$-sphere

I know this probably seems like a dumb question, I have parametrised part of the unit $3$-sphere with $(x,y,z)\to (x,y,z,(1-(x^2+y^2+z^2))^{\frac{1}{2}})$ and now I'm trying to calculate the ...
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1answer
28 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
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Representation of complex Clifford algebra on exterior algebras when quadratic form has odd index

Overview This problem entails the explicit construction of representation of Clifford algebra upon the exterior algebra, using orthogonal complex structure or polarization, namely, given a ...
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32 views

Parallel $ (1,1) $-forms on compact Kähler manifolds.

Let $ (X,\omega) $ be a compact Kähler manifold. We know one example of a parallel $ (1,1) $-form, namely, $ \omega $ itself. Are there obstructions for the existence of non-vanishing parallel $ (1,1) ...
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1answer
30 views

Closed geodesic minimizing properties

Considering closed geodesics on a compact manifold M of even dimension, what does it mean to say that a curve (any closed geodesic) is locally energy minimizing but not globally ? For simplicity, say ...
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Closed geodesics on real projective space

We have the result that all closed geodesics on $S^n$ must be contained with the intersection of $S^n$ and a plane. Hence all length minimising closed geodesics are single points. If we equip ...
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31 views

Green's function for Laplace operator in a conformally flat metric?

Given the Laplace–Beltrami operator $\nabla^2$, does there exists a closed form for the greens function $G$ such that $\nabla_x^2G(x,y)=-\delta(x,y)$, and $$ \nabla_x^2\iiint_{y^3}G(x,y)f(y)dy^3=-f(x) ...
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2answers
37 views

Expressing a metric as a sum of (possibly) many squares

Given a Riemannian manifold $M$ whose metric $g$ has zero curvature, it is known that we can find local coordinates $x^i$ such that $$g=\sum_{i=1}^{\dim(M)}(dx^i)^2.$$ Conversely, if the curvature ...
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1answer
85 views

Is $S^1 \times S^1$ really a torus?

Consider a function $f(x)$ that is $2\pi$ periodic. Consider another function $g(y)$ that is also $2\pi$ periodic. If I wanted to compute the integral of either of these functions I would do so ...
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1answer
93 views

What is the smallest Euclidean space in which one can embed a given curved space?

Given a $d$-dimensional curved space, how many dimensions are required to embed it? As an example think of a sphere's surface, which is a two-dimensional curved space that can be expressed in ...
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1answer
27 views

Mean value theorem on Riemannian manifold?

Is there some generalisation of the classical mean value theorem for real-valued functions on an interval $$|f(x)-f(y)| \leq |\nabla f(c)||x-y|$$ for some $c$ between $(x,y)$ to the case where $f:M ...
2
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2answers
43 views

The relationship between Ricci and Gaussian curvatures

Why do we have that for a surface (dimension $2$) that $$\text{Ric}(X, Y) = K \langle X, Y \rangle ,$$ where $K$ is the Gaussian curvature and $X, Y$ are vector fields?
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1answer
74 views

Relation between parallel vector field along a geodesic and Jacobi field along that same geodesic

Cross posted from my question: http://mathoverflow.net/questions/204097/parallel-transport-along-a-geodesic-and-the-related-jacobi-field This is a formula/theorem (written below) that I found ...
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1answer
36 views

What is a local invariant?

Let $(M,g)$ be a Riemannian manifold. Then, it is usually said that $M$ has local invariants associated to $g$. For example, the curvature of the Levi-Civita connection associated to $g$. My question ...
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General setting of Varadhan's result for distance functions and heat kernels

For a senior project of mine, I would like to know what the most general setting of Varadhan's formula for the geodesic distance in terms of the limiting behavior of heat kernels is. The result I'm ...
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50 views

Intuitive interpretation of Ricci Flow

What is the best way to interpret, explain or somehow visualize the basic idea behind formal definition of Ricci Flow? I am familiar with the hackneyed expressions like "Ricci Flow is a ...
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Mathematica packages for symbolic computations of Christoffel symbols and parallel transports in Riemannian geometry, [closed]

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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1answer
53 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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1answer
48 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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1answer
29 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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1answer
27 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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2answers
33 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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1answer
58 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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25 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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1answer
49 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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1answer
37 views

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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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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3answers
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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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1answer
42 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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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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36 views

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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1answer
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
35 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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1answer
63 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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1answer
43 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
40 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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1answer
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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
20 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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44 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
50 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 ...