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Here is an example: Let $\gamma:\mathbb{R}\to\mathbb{R}^3$ be defined by $\gamma(t) = \bigl(t,\exp(-1/t^2),0\bigr)$ for $t<0$, $\gamma(0) = (0,0,0)$ and $\gamma(t) = \bigl(t,0,\exp(-1/t^2)\bigr)$ for …

As I suspected, the statement that the bundle $\mathrm{SO}(8)\to S^7$ is a product bundle, i.e., that $$ \mathrm{SO}(8)\simeq S^7\times\mathrm{SO}(7)\tag1 $$ as bundles over $S^7$ is in N. Steenrod's …

If you don't care about completeness, here's a fairly simple way to construct such examples: Start with a compact Einstein manifold $(M^n,g)$ with Einstein constant $1$ (i.e., $\mathrm{Ric}(g) = (n{- …

First, the definition: A Riemannian $n$-manifold $(M^n,g)$ is of Hessian type if there exist $(n{+}1)$ functions $x^1,\ldots,x^n, u$ on $M$ such that $dx^1\wedge\cdots\wedge dx^n\not=0$ and such that …

There are local conditions, but they typically involve the curvature tensor of the underlying metric. For example, if the metric is flat, so that one can choose orthonormal coordinates $x_i$ in which …

One then finds that the integral manifolds of the differential system must satisfy $$ \begin{align} v_5 &= -\sinh(\rho)\ (p_4{+}q_5)-\cosh(\rho)\ (p_5{+}q_4)\\\\ v_6 &= (\cosh^2(\rho){-}\cosh(\rho)\sinh … Thus, with this construction and the above formulae, we have the complete description of the (local and global) integral manifolds of the original EDS in terms of the second and third derivatives of $L …

NB: As the OP pointed out, the first version of my answer was incomplete, since it didn't address the 'tubular neighborhood' part of the claim. Here's a better (but still not complete) version of an …

For $\mathrm{SU}(2)$, with the scale for the biïnvariant metric so that it becomes isometric to the unit $3$-sphere $S^3$ in Euclidean $4$-space, it is well-known what the eigenvalues of the Laplace-B …