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Let $F$ generate a nonsingular smooth flow on a closed manifold. Suppose a continuous nondegenerate indefinite quadratic form on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative along that flow. Then the flow is an Anosov flow. Compactness makes the positivity uniform; the positive and negative cones give the unstable and stable transverse directions. The flow direction must be quotiented out: its derivative cannot satisfy strict positivity.

If a connected closed manifold $M^3$ has a noncontractible universal cover, then $M\times \mathbb{R}$ has no complete metric making it a Ricci-flat Riemannian manifold. The product has two ends and hence contains a line in a Riemannian manifold for any complete metric. The Cheeger-Gromoll splitting theorem would split it isometrically as $N^3\times \mathbb{R}$. Its three-dimensional factor is Ricci-flat and therefore flat by three-dimensional curvature from the Ricci tensor. The complete simply connected cover of the product would be Euclidean four-space. But that cover is $\tilde{M}\times \mathbb{R}$, which retracts onto $\tilde{M}$, contradicting noncontractibility.

For the positive Laplace-Beltrami operator, the kernel solving $({\partial }_t+{\Delta }_x)H=0$ with initial Dirac delta distribution on the diagonal represents the heat semigroup. On a closed manifold with a Riemannian metric it is smooth for $t>0$, symmetric, nonnegative, preserves constants, and obeys the semigroup property.

For a metric vector bundle connection over a Riemannian manifold, the rough Laplacian is the formally nonnegative connection operatorwhere $e_i$ is a local orthonormal frame. The correction term makes the expression independent of the orthonormal frame. On a closed manifold, integration by parts gives $⟨{\nabla }^{\ast }\nabla s,s{⟩}_{L^2}=\Vert \nabla s{\Vert }_{L^2}^2$.

For positive $p,q$, quotient $S^p\times S^q$ by $(x,y)\mapsto (-x,-y)$. This is a free double covering space action. Its closed manifold quotient is orientable exactly when $p+q$ is even, because the product mapping degree is $(-1{)}^{p+q+2}$. If $p,q\ge 2$, the product is simply connected and the quotient has fundamental group $\mathbb{Z}/2$. For $p=q=2$, the quotient has integral cohomology groups $\mathbb{Z}$ in degrees $0,4$, $\mathbb{Z}/2$ in degrees $2,3$, and zero elsewhere. Every positive-degree cup product is zero. These conclusions follow from Poincare duality, the Euler characteristic under a finite covering, and the universal coefficient theorem for cohomology.

A smooth flow is a smooth one-parameter family of diffeomorphisms satisfying ${\varphi }_0=\mathrm{id}$ and ${\varphi }_{t+s}={\varphi }_t\circ {\varphi }_s$. Its generating vector field is $F(x)={\partial }_t{\varphi }_t(x){|}_{t=0}$. On a closed manifold, every smooth vector field has a flow defined for all real times.

On a closed manifold with a Riemannian metric, an orthonormal eigenbasis for the positive Laplace-Beltrami operator gives this formula, with eigenvalues counted with multiplicity. Polynomial elliptic bounds and exponential time decay give convergence of every derivative away from time zero. The conjugate is necessary for a complex basis.

For a nonnegative elliptic operator $L$ on a closed manifold, sum its positive eigenvalues with multiplicity. The defining series converges in a right half-plane. If $n_0=\dim \ker L$ and $K(t)=\mathrm{Tr}(e^{-tL})$ is the heat trace, thenSubtracting finitely many terms of the short-time heat kernel expansion gives a meromorphic continuation. Zero eigenvalues must be excluded: assigning them a nonexistent inverse power does not define the spectral zeta function.

For a finite subgroup $U$ of a group $T$ of Riemannian isometries of a closed manifold $N$, assume $U$ acts freely and define $I_t(g)={\int }_N{H}_N(t,x,gx)dV(x)$. Invariance of the Riemannian heat kernel and change of variables show that $I_t$ is a class function on $T$. For $U\le T$, the heat kernel on a finite isometric quotient and integration over a fundamental domain give the displayed heat trace formula. Grouping by conjugacy classes proves equality for Gassmann equivalent subgroups. More generally the same argument allows fixed points of $T$ outside the freely acting subgroups; only the quotient under consideration must be smooth.

On a spin manifold, tensor the spinor bundle $S$ with a Hermitian vector bundle $W$ carrying a unitary connection. Composing the induced tensor-product connection on a vector bundle with Clifford multiplication defines the twisted Dirac operator. In even dimension it maps $S^{+}\otimes W$ to $S^{-}\otimes W$ and conversely. On a closed manifold its chiral restriction $D_W^{+}:H^1(S^{+}\otimes W)\to L^2(S^{-}\otimes W)$ is a Fredholm operator.

The universal cover of a connected closed manifold of dimension three with finite fundamental group is a closed simply connected three-manifold. It is orientable, since its orientation character has trivial domain. Its first integral homology vanishes, and the universal coefficient theorem for cohomology then gives $H^1=0$. Integral Poincare duality gives $H_2=H^1=0$, while $H_0$ and $H_3$ are $\mathbb{Z}$. Thus it is an integral homology sphere. This proof does not require classifying simply connected three-manifolds.

If $R=({\partial }_t+{\Delta }_x)P$ is smooth and bounded up to time zero on a closed manifold, the Volterra convolution of kernels solves $Q+R+R\ast Q=0$ by $Q={\sum }_{k\ge 1}(-1{)}^k{R}^{\ast k}$. Bounds by $C^k\mathrm{vol}(M{)}^{k-1}{t}^{k-1}/(k-1)!$ prove convergence. Then $H=P+P\ast Q$ is the exact Riemannian heat kernel and has the same initial delta limit.

For the positive Laplace-Beltrami operator $P$ on a closed manifold, the wave trace is the distribution ${\sum }_j{e}^{it\sqrt{{\lambda }_j}}$, counting multiplicities. The series is interpreted after testing against smooth compactly supported functions, rather than as an ordinarily convergent sum. Its singularities relate the spectrum to the length spectrum of closed geodesics, through the Duistermaat-Guillemin trace formula.

For a Laplace-type operator on a closed manifold, let $a_{t^0}$ be the constant coefficient of its full heat trace expansion. In the Mellin transform defining the spectral zeta function, the reduced trace has constant coefficient $a_{t^0}-\dim \ker L$. Its contribution is $(a_{t^0}-\dim \ker L)/s$, and $1/\Gamma (s)=s+O(s^2)$ proves the identity. For the scalar positive Laplace-Beltrami operator on a closed surface, $a_{t^0}=\int RdV/(24\pi )$ and $\dim \ker L$ is the number of connected components. In particular, a connected sphere has value $-2/3$ and a connected flat torus has value $-1$.