# Tagged Questions

Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

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### Dimension of tangent space to manifold of cross section slices

Given a function $\Phi:\Omega^{\Phi}\subset \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\Omega^{\phi}\subset \mathbb{R}^2\rightarrow\mathbb{R}$, using a ...
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### On the curvature tensor with certain conditions

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\lbrace X_1,...,X_n,Y_1,...,Y_m\rbrace$ be a locally orthonormal frame for $M$($3\leq n,m$). If we suppose the curvature tensor $R$ of $g$ ...
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### Existence of left-invariant metric on the cotangentbundle of homogeneous spaces?

Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ ...
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### Trace Theorem for $q< 2(n-1)/(n-2)$

Can I get a trace theorem inequalite for $R^n_+$: For $q\in [2,2(n-1)/(n-2]$, we have $(\int_ {R^{n-1} } |u|^q dx) ^{2/q}\leq C(\int_{R^n_+} |\nabla u |^2dx)^{1/2}.$
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### Scattering in (pseudo-)Riemannian spaces

I will ask my question in a broad way, leaving a lot of freedom for answers. Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
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### Local differential geometry and invariant theory

Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists? Start with ...
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### Parallel Transport on Hypersurface Spinor Bundle

So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link: https://projecteuclid.org/...
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### Euclidean metric in spherical coordinates [closed]

I apologize if this question is not research level, but it has been asked on MSE before, and not received really satisfactory answers, for example, see here, and googling does not reveal anything ...
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### Hodge map and the Cohomology Ring of a Riemannian Manifold

For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology ...
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### Obtaining Hessian of the embedding from an induced metric

Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...
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### Flows associated with Killing fields

Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
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### Upper bound for Willmore energy

Good day to everyone! Does anybody know if there are upper bound estimates for Willmore energy for a given surface?
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### material derivative and relation to Riemannian metric

For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$. Smooth functions on $N$...
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### Differentiability of a map to the free loop space

While reading Morse theory, closed geodesics, and the homology of free loop spaces, the author claims the following: Given the $S^{n-1} \hookrightarrow Y_1 \rightarrow T^1S^n$ bundle over $T^1S^n$, ...