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Since I'm lazy, I'm shamelessly referring to the following question: Derivative of Exponential Map Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector …

Also asked here: https://math.stackexchange.com/questions/1725787/when-are-the-minimizing-geodesics-of-a-totally-geodesic-submanifold-also-minimiz A reference on totally geodesic submanifold (TGS): …

Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com, https://math.stackexchange.com/quest …

I asked this question on math.stackexchange too: it's not a homework problem, but something that came to my mind while thinking of commutation: https://math.stackexchange.com/questions/1356518/differ …

Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$, and $exp:\mathfrak{g}\to G$ be its exponential map. The group $G$ could be finite or infinite dimensional. Let $G$ have the property that $\ …

I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them …

Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$. QUESTION I: The above s …

Let $M \subset \mathbb{R}^p$ be a Riemannian submanifold. In what follows, when we talk about a Lipschitz function $f$ on $M$, namely $f: M \to \mathbb{R}$, we will assume there is a $L > 0$ so that: …

I apologize if this is not a research level question (already tried asking https://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange wit …

Also asked here: https://math.stackexchange.com/questions/1725491/riemannian-metric-on-a-level-set-of-a-smooth-function-on-a-manifold Let $(M,g)$ be a finite or infinite dimensional Riemannian manifo …

Intuitively it might be clear that differential geometry is a very applicable subject in engineering and industry. I'd like to know how some industries/companies use differential geometry. I'd guess t …

Setup: I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes. Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{* …

Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra …

We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\ …

My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer: Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a discret …