# Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

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### Definition of static spherically symmetric spacetime as fiber bundle

I am working on a physical paper about solutions of Einstein field equations in case of static spacetimes with perfect fluid spheres and wanted to invent a new definition of spherical symmetry there. ...
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### Show the indefinite unitary group acts transitively on the hyperbloid manifold

Consider an indefinite Hermitian form $\langle \cdot ,\cdot \rangle$ on $\mathbb{C}^{n+1}$ such that $$\langle v ,w \rangle = \sum^n_{i=1} v_i \bar{w}_i - v_{n+1} \bar{w}_{n+1}.$$ We let $U(n,1)$ ...
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### Calculating the product of the Riemannian manifolds and the Riemann curvature [closed]

I am a physics Master student currently taking a course on Riemannian geometry. In the course we are supposed to solve problem 7-4 out of Lee's Book Introduction to Riemannian Manifolds. The Problem ...
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### How to calculate the volume of the image of the manifold

Let $M$ be a $n$ dimensional manifolds, $f:M \rightarrow \mathbb {R}^n$ be a smooth map. Then, how can I calculate $\textrm{vol}(fM)$ ? I'm thinking of calculating it using the area formula as shown ...
27 views

### What is the precise definition of the Darboux tangent to a surface? [closed]

What is the definition of Darboux tangents of a surfaces? The book "Affine Differential Geometry" mentions it, but it does not give a precise definition.
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### Properties of the linearized Riemann tensor?

The linearized Riemann tensor is given by: $$R_{\alpha \beta \mu \nu}=-\frac{1}{2}\left[h_{\alpha \mu, \beta \nu}+h_{\beta \nu, \alpha \mu}-h_{\alpha \nu, \beta \mu}-h_{\beta \mu, \alpha \nu}\right]$$ ...
1 vote
19 views

### Unique representation of $\mathbf{Gr}^+(p,n)$ the oriented real Grassmannian

For $\mathbf{Gr}(p,n)$ the $p$ dimensional subspace of $\mathbb{R}^n$, or equivalently $O(n)/O(p)\times O(n-p)$, a point has a unique projector representation $P = UU’$ where U is an $n \times p$ ...
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### Is the map $T_X |_S (p) := \exp{X(p)}$ a diffeomorphism onto its image?

Preliminaries The exponential map $\exp : TM \rightarrow M$ is defined by $\exp{(v)} = \gamma_v (1)$ where $\gamma_v$ denotes the geodesic starting at $p \in M$ and initial velocity $v \in T_p M$. ...
1 vote
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### Conjugate points and expansion of the geodesic congruence

I am working in a Lorentzian manifold $(M, g)$ (but I think the problem would be quite similar in a Riemannian manifold) and I am considering a timelike geodesic whose tangent vector field is denoted ...
58 views

### Calculating the gradient of Log-Euclidean distance between SPD matrices on Riemannian manifold

In the paper Log-Euclidean metrics for fast and simple calculus on diffusion tensors, the geodesic distance between SPD matrices $A,B$ is defined as $$d(A,B)=||\log A- \log B||_F,$$ where $F$ is the ...