Tagged Questions

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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Showing that Ricci curvature of round unit sphere $(S^n,g_0)$ is $Ric(g_0)=(n-1)g_0$

Let $g_0$ is a Riemannian metric of round unit sphere, $Ric(g_0)$ is the Ricci curvature. How to show that the $Ric(g_0)$ of round unit sphere $(S^n,g_0)$ is $Ric(g_0)=(n-1)g_0$ ? In fact, I don't ...
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Does any minimal surface which is regular has a conformal parameterization?

Since given surface is regular and minimal, I have $$H=\frac{Eg-2Ff+Ge}{2(EG-F^2)}=0$$ and $$X_u\times X_v\ne0$$ Can I derive $E=G,\ F=0$ from these conditions?
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Extend a Riemannian metric defined on the boundary.

If $M$ is a (compact) Riemannian manifold with nonempty boundary and I have a Riemannian metric defined on $\partial M$, Is it possible to obtain a Riemannian metric on the whole $M$ extending the one ...
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Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
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The Hessian at a critial point $p$

In my studies I've come across the hessian in the context of Riemannian geometry. I use the following definition of the hessian $$H^f(X,Y)=XYf-(D_xY)f=\langle D_X(\operatorname{grad} f),T\rangle.$$ ...
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Divergence equation on noncompact manifold.

Is it true that on any complete, non-compact Riemannian manifold $(M, g)$ there is a smooth vector field X such that $\operatorname{div} X=1$? My definition of the divergence is the coordinate one (...
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Riemmaninan Metric Outer measure?

If $M$ is a Riemmanian manifolds and $\mu$ is the Riemmanian measure thereon, then is $\mu$ equal to the metric-outer measure induced $M$'s Riemmanian metric when it is restricted to Borel sets?
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complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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Geodesics that paint curves on the projective space bundle

Let $M$ be a geodesically complete connected riemannian manifold. Let $p \in M$ be a point and $c: \mathbb{R} \to M$ an arbitrary geodesic that doesn't intersect p. Our aim is to find a "nice" map ...
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Tensor Algebra for Riemannian Geometry

I'm trying to learn a little bit about Riemannian geometry, but the books that I'm looking through seem to assume that the reader is familiar with topics such as contracting tensors, raising and ...
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Bundle of densities

On the orientable manifold $M$ one can find a nowhere vanishing top form $\omega$: in fact the existence of such a form is equivalent to $M$ being orientable. In the nonorientable setting it is ...
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Metric compatibility, Ricci rotation coefficients & non-coordinate bases

I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric ...
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construct a connection such that a given tensor is parallel wrt it

Let $\omega$ be a symplectic form on a smooth manifold $M$. How does one construct a connection on $TM$ such that $\omega$ is parallel to it? It's easy to construct a connection on a dual bundle ...
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Linear transformation and riemannian metric tensor

When is it true that for a linear transformation $A$ and a symmetric bi-linear form $g$ (i.e. (0,2) tensor) on a riemannian manifold: $$g( AV, W)= g(V, AW)?$$
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My question is in relation to the derivation of the formal adjoint for a connection $D:\Omega^{p-1}(\text{Ad}E)\rightarrow \Omega^p(\text{Ad}E)$ - I am reading through this derivation in Jost's ...
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Show that $\overline{R}(X,Y)Z=\nabla_X\nabla_YZ-\nabla_{[X,Y]}Z=-\Theta$.

Exercise I am tasked with showing the following (Exercise 2.6.13.4 in Cartan For Beginners by Ivey and Landsberg): For $X,Y,Z\in\Gamma(TM)$, define  \overline{R}(X,Y)Z=\nabla_X\nabla_YZ-\...
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What's the lemma's name ?And how to proof it?

I saw the lemma in a Riemannian Geometry book, but the proof is omitted ,I want to know how to proof it .Thanks.
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Are two isometric Riemannian manifolds necessarily related by Euclidean motions?

Let $(M,g), (\overline{M},\overline{g})$ be smooth Riemannian manifolds that are isometric, i.e. there is a smooth function $f: M \rightarrow \overline{M}$ such that d$f(g)=\overline{g}$. By Nash's ...
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Integral on Riemannian manifold

$M$ is a Riemannian manifold ,and $g_{ij}$ is Remannian metric. Let $x=(x^1...x^d)$ $(i.e. x:U_x\rightarrow R^d)$ be a local coordinates ,and $v,w\in T_pM$ with coordinate representations $(v^1...v^d)$...
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Fixing some notations about tensor calculus

I'm studing some articles about Ricci Flow and Ricci Solitons. I realize the first thing to do is choose a notation, thus I'm using the Huai Dong Cao's notation (which is the same Ricgard Hamilton's ...
I am reading these notes on differential geometry from a course at MIT. I have been verifying the computations for myself and I have a concern about the expression for $de^k$ on the final line. When I ...