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A rectifiable curve (in the sense of, for example, baby Rudin, p. 136) in Euclidean space or a metric space more generally, is precisely a continuous curve with finite length.

Question 1: Domain and target; the $\nabla$ takes $\Omega(M,E)$ to itself. Question 2: Property (*) defines the extension on pure wedge products $\omega \wedge \sigma$, but every element of $\Omega(M …

You are looking for the theory of projective connections. See the paper of Molzon and Mortensen, The Schwarzian derivative for maps between manifolds with complex projective connections, Trans. Amer. …

Here is the main idea. Suppose that the metric extends smoothly to the boundary. Such a metric has two null directions at each point. Follow one until you reach the boundary, and then switch to the ot …

If we take a basis $\sigma_A$ of local sections of $E$, any connection on $E$ is expressed as $\nabla \sigma_A = \gamma^B_A \sigma_B$, where $\gamma^B_A$ are the connection 1-forms. On $E^*$, the indu …

I am fairly certain that no one has considered pullbacks of Lie algebra cocycles in the context of Cartan geometries.

It is not always possible, for example you cannot extend the unit normal vector to the 2-sphere to a nonzero vector field inside the ball. I gave this problem on my 3rd year undergraduate analysis exa …

Always: the definition of $\Delta_g$ is in terms of $g$, so any diffeomorphism preserving $g$ preserves $\Delta_g$, and this is what ``preserves'' means.

If $H$ is a reductive group (for example, compact or semisimple), then the representation $\mathfrak{g}$ splits over $H$, and you do indeed get an induced connection, by splitting $\omega$ into its pa …

If $K$ is to be the Gauss curvature of some metric on a compact two-dimensional manifold $M$ with Euler characteristic $χ(M)$, then from the Gauss-Bonnet theorem one has the following conditions: (a) …

If two vector bundles $V_1$ and $V_2$ have connections $\nabla_1$ and $\nabla_2$, then the vector bundle $V=V_1 \otimes V_2$ has a unique connection $\nabla$ for which $\nabla(s \otimes t)=(\nabla_1 s …

There is no standard notation. The problem is simply the noncommutativity, making the expressions unwieldy. It is rare in geometry to differentiate many times using a connection. Twice is usually enou …

There is a theory due to Tresse, which is unfortunately fairly complicated. It is explained, at least partly, in the book of Arnol'd, Geometric Methods in the Theory of Ordinary Differential Equations …

Correction: The following idea doesn't work as stated, because (as Robert points out) the cusp and cones allow points where mean curvature is not defined. If you could isometrically and smoothly embed …

I think you are probably looking to find an analogue of the Serret--Frenet theory for higher dimensional submanifolds of Euclidean space; this is discussed in detail in What is the analog of the "Fund …