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Proposition (Bochner-weitzenbock formula). For any $p$-form $\beta$, $$\frac{1}{2}\Delta|\beta|^2=|\nabla \beta|^2-\langle\Delta \beta,\beta\rangle+F(\beta),$$ where $F$ is a complicate expression related to curvature tensor. For proof and more information see Poor, Walter A., Differential geometric structures, New York etc.: McGraw-Hill Book Company. XIII, ...


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Let $M=\mathbb{R}$. Then there is an effective action by the $2$-dimensional group of affine maps $\{x\mapsto ax+b|a\in \mathbb{R}\backslash\{0\},\,\,b\in \mathbb{R}\}$. Also another example, where $M$ is compact: $PSL_2(\mathbb{R})$ is $3$-dimensional and acts effectively on $S^1=\mathbb{R}P^1$ by Mobius transforms.


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There are a number of problems with your argument. As @Didier commented, it does not follow from the absence of conjugate points that $\gamma$ is length-minimizing. More importantly, most of your claims about $I(V,V)$ and $I(J,J)$ would be justified if $V$ and $J$ were proper (i.e., vanishing at the endpoints), but there's no such assumption in the problem ...


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The points $p, q, \gamma_1(t), \gamma_2(t)$ span a Saccheri quadrilateral $Q$ in the hyperbolic plane, where $d$ is the base and $t$ is the common length of the legs of $Q$. The hyperbolic distance between $\gamma_1(t), \gamma_2(t)$ is called the summit $s$ of $Q$. The correct formula for the summit is not $d\cosh(t)$ but $$ \cosh(s)= \cosh(d) \cosh^2(t) - \...


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Here's one longgggg not actually very simple sentence, that a good second year undergraduate STEM student who knows multivariable calculus might possibly maybe be able to understand: While one ordinarily computes area in $\mathbb R^2$ by a double integral $$\int\!\!\int dx \, dy $$ or computes path length by a path integral $$\int \sqrt{\frac{dx}{dt}^2 + \...


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