# Questions tagged [riemannian-geometry]

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

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### Taylor expansion on a Manifold

Is there a way to define the Taylor expansion of a function $f:\mathcal{M}\rightarrow\mathbb{R}$, where $\mathcal{M}$ is a smooth manifold? I'm looking for a free coordinate definition. I guess it is ...
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### Lemma 4.14 Heat Kernels and Dirac Operators

I am trying to work out Lemma 4.14 of the book "Heat Kernels and Dirac Operators" by Berline and Getzler. I am stuck with the proof. For the sake of brevity I am uploading a picture from the book ...
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### Relation between the differential of a vector field and the covariant derivative

Let $\mathcal{M}$ be a Riemannian manifold and let $X:\mathcal{M} \rightarrow \mathcal{T}\mathcal{M}$ be a smooth vector field, i.e., $X(p) \in \mathcal{T}_p\mathcal{M}$ $\forall p \in \mathcal{M}$. ...
I have a Riemannian metric on $R^3$ whose matrix is written as $h=P^TDP$, where $P\in SO(3)$ and $D$ is a diagonal matrix with positive valued smooth functions on the diagonal. Here $SO(3)$ is the ...
Let $(M^3,g)$ be a compact Riemannian manifold with boundary $\partial M \neq \emptyset$, and consider a non constant harmonic function $f : M \to \mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$ satisfying ...